Geometric Algebra

Geometric algebra From Wikipedia, the free encyclopedia

Contents 1

*-algebra 1.1

2

3

4

5

1

Terminology

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1

1.1.1

*-ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.2

*-algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.3

*-operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.4

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3

Additional structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3.1

Skew structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.4

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.5

Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

Acceptable ring

5

2.1

5

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Additive identity

6

3.1

Elementary examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

3.2

Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

3.3

Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

3.4

Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3.4.1

The additive identity is unique in a group . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3.4.2

The additive identity annihilates ring elements . . . . . . . . . . . . . . . . . . . . . . . .

7

3.4.3

The additive and multiplicative identities are different in a non-trivial ring . . . . . . . . . .

7

3.5

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3.6

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3.7

External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

Additive map

9

4.1

Additive map of a division ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

4.2

References

9

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Algebra (ring theory)

10

5.1

Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

5.2

Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

i

ii

CONTENTS 5.2.1

6

7

Split-biquaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

5.3

Associative algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

5.4

Non-associative algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

5.5

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

5.6

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

5.7

Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

Algebra homomorphism

13

6.1

Unital algebra homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

6.2

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

6.3

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

6.4

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

Algebra of physical space

15

7.1

Special relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

7.1.1

Space-time position paravector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

7.1.2

Lorentz transformations and rotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

7.1.3

Four-velocity paravector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

7.1.4

Four-momentum paravector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

Classical electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

7.2.1

The electromagnetic field, potential and current . . . . . . . . . . . . . . . . . . . . . . .

17

7.2.2

Maxwell’s equations and the Lorentz force . . . . . . . . . . . . . . . . . . . . . . . . . .

18

7.2.3

Electromagnetic Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

7.3

Relativistic quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

7.4

Classical spinor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

7.5

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

7.6

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

7.6.1

Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

7.6.2

Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

7.2

8

9

Algebra representation

20

8.1

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

8.1.1

Linear complex structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

8.1.2

Polynomial algebras

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20

8.2

Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

8.3

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

8.4

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

8.5

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

Algebraically compact module

22

9.1

Definitions

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22

9.2

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

9.3

Facts

23

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CONTENTS

iii

9.4

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

9.5

References

23

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10 Almost commutative ring

24

10.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

10.2 References

24

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11 Annihilator (ring theory)

25

11.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

11.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

11.3 Chain conditions on annihilator ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

11.4 Category-theoretic description for commutative rings . . . . . . . . . . . . . . . . . . . . . . . . .

26

11.5 Relations to other properties of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

11.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

11.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

11.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

12 Arithmetical ring

28

12.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

12.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

13 Artin algebra

29

13.1 Dual and transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

13.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

14 Artinian ideal

30

14.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

14.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

15 Artinian module

31

15.1 Left and right Artinian rings, modules and bimodules . . . . . . . . . . . . . . . . . . . . . . . . .

31

15.2 Relation to the Noetherian condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

15.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

15.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

15.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

16 Artinian ring

33

16.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

16.2 Modules over Artinian rings

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33

16.3 Commutative Artinian rings

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34

16.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

16.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

16.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

16.4 Simple Artinian ring

iv

CONTENTS

17 Artin–Rees lemma

36

17.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

17.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

17.3 Proof of Krull’s intersection theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

17.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

17.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

18 Artin–Wedderburn theorem

38

18.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

18.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

18.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

19 Artin–Zorn theorem

40

19.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Ascending chain condition on principal ideals

40 41

20.1 Commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

20.2 Noncommutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

20.3 References

42

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21 Associated graded ring

43

21.1 Basic definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

21.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

21.3 Generalization to multiplicative filtrations

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

21.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

21.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

22 Associated prime

45

22.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

22.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

22.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

22.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

23 Augmentation ideal 23.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Azumaya algebra 24.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Baer ring

48 49 50 50 51

25.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

25.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

25.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

25.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

CONTENTS

v

25.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

25.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

26 Balanced module

53

26.1 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

26.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

26.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

27 Beauville–Laszlo theorem

55

27.1 The theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

27.2 Global version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

27.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

28 Bimodule

57

28.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

28.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

28.3 Further notions and facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

28.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

28.5 References

58

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29 Binomial ring

59

29.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Biquaternion

59 60

30.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

30.2 Place in ring theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

30.2.1 Linear representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

30.2.2 Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

30.3 Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

30.3.1 Relation to Lorentz transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

30.4 Associated terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

30.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

30.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

30.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

31 Bivector

65

31.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

31.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

31.2.1 Geometric algebra and the geometric product . . . . . . . . . . . . . . . . . . . . . . . .

66

31.2.2 The interior product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

31.2.3 The exterior product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

31.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

31.3.1 The space Λ2 ℝn

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

31.3.2 The even subalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

vi

CONTENTS 31.3.3 Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

31.3.4 Unit bivectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

31.3.5 Simple bivectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

31.3.6 Product of two bivectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

31.4 Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

31.4.1 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

31.5 Three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

31.5.1 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

31.5.2 Rotation vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

31.5.3 Rotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

31.5.4 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

31.5.5 Axial vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

31.5.6 Geometric interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

31.6 Four dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

31.6.1 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

31.6.2 Simple bivectors in 4D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

31.6.3 Rotations in ℝ

4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

31.6.4 Spacetime rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

31.6.5 Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

31.7 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

31.7.1 Rotations in higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

31.8 Projective geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

31.8.1 Tensors and matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

31.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

31.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

31.11General references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

32 Blade (geometry)

86

32.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

32.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

32.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

32.4 General references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

32.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

33 Boolean ring

88

33.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

33.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

33.3 Relation to Boolean algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

33.4 Properties of Boolean rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

33.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

33.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

33.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

CONTENTS

vii

34 Brauer group

91

34.1 Construction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

34.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

34.3 Brauer group and class field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

34.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

34.5 General theory

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

34.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

34.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

34.8 References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

34.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

34.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

35 Buchsbaum ring

95

35.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Bézout domain

95 96

36.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

36.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

36.3 Modules over a Bézout domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

36.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

36.5 References

98

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 C-semiring 38 Cartan–Brauer–Hua theorem

99 100

38.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 39 Category of rings

101

39.1 As a concrete category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 39.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 39.2.1 Limits and colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 39.2.2 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 39.2.3 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 39.3 Subcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 39.3.1 Category of commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 39.3.2 Category of fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 39.4 Related categories and functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 39.4.1 Category of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 39.4.2 R-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 39.4.3 Rings without identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 39.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 40 Central polynomial

105

40.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

viii

CONTENTS 40.2 References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

41 Central simple algebra

106

41.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 41.2 Splitting field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 41.3 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 41.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 41.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 41.5.1 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 42 Centralizer and normalizer

109

42.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 42.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 42.2.1 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 42.2.2 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 42.2.3 Rings and algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 42.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 42.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 42.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 43 Change of rings

112

43.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 43.2 References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

44 Characteristic (algebra) 44.1 Other equivalent characterizations

113 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

44.2 Case of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 44.3 Case of fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 44.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 45 Classification of Clifford algebras

116

45.1 Notation and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 45.2 Bott periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 45.3 Complex case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 45.4 Real case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 45.4.1 Classification of quadratic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 45.4.2 Unit pseudoscalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 45.4.3 Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 45.4.4 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 45.4.5 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 45.5 Bibliography

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

45.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 46 Clifford algebra

120

CONTENTS

ix

46.1 Introduction and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 46.1.1 As a quantization of the exterior algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 46.2 Universal property and construction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

46.3 Basis and dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 46.4 Examples: real and complex Clifford algebras

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

46.4.1 Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 46.4.2 Complex numbers

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

46.5 Examples: constructing quaternions and dual quaternions . . . . . . . . . . . . . . . . . . . . . . . 123 46.5.1 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 46.5.2 Dual quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 46.6 Examples: in small dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 46.6.1 Rank 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 46.6.2 Rank 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 46.7 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 46.7.1 Relation to the exterior algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 46.7.2 Grading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 46.7.3 Antiautomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 46.7.4 Clifford scalar product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 46.8 Structure of Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 46.9 Clifford group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 46.9.1 Spinor norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 46.10Spin and Pin groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 46.11Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 46.11.1 Real spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 46.12Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 46.12.1 Differential geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 46.12.2 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 46.12.3 Computer vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 46.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 46.14Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 46.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 46.16Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 46.17External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 47 Coherent ring

136

47.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 48 Commutative ring

137

48.1 Definition and first examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 48.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 48.1.2 First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 48.2 Ideals and the spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

x

CONTENTS 48.2.1 Ideals and factor rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 48.2.2 Localizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 48.2.3 Prime ideals and the spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 48.3 Ring homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 48.4 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 48.5 Noetherian rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 48.6 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 48.7 Constructing commutative rings

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

48.7.1 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 48.8 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 48.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 48.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 48.10.1 Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 48.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 49 Comodule

144

49.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 49.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 49.3 Rational comodule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 49.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 50 Comparison of vector algebra and geometric algebra 50.1 Basic concepts and operations

146

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

50.2 Embellishments, ad hoc techniques, and tricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 50.3 List of analogous formulas

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

50.3.1 Algebraic and geometric properties of cross and wedge products

. . . . . . . . . . . . . . 147

50.3.2 Norm of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 50.3.3 Lagrange identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 50.3.4 Determinant expansion of cross and wedge products . . . . . . . . . . . . . . . . . . . . . 148 50.3.5 Matrix Related . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 50.3.6 Equation of a plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 50.3.7 Projection and rejection

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

50.3.8 Area of the parallelogram defined by u and v

. . . . . . . . . . . . . . . . . . . . . . . . 154

50.3.9 Angle between two vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 50.3.10 Volume of the parallelopiped formed by three vectors . . . . . . . . . . . . . . . . . . . . 155 50.3.11 Derivative of a unit vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 50.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 51 Composition algebra

157

51.1 Structure theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 51.2 The case char(K) ≠ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 51.2.1 Scalar product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

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51.2.2 Involution in Hurwitz algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 51.2.3 Para-Hurwitz algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 51.2.4 Euclidean Hurwitz algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 51.3 Instances and usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 51.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 51.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 52 Composition ring

161

52.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 52.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 52.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 53 Composition series

163

53.1 For groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 53.1.1 Uniqueness: Jordan–Hölder theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 53.2 For modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 53.3 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 53.4 For objects in an abelian category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 53.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 53.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 53.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 54 Conformal geometric algebra

166

54.1 Construction of CGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 54.1.1 Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 54.1.2 Base and representation spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 54.1.3 Mapping between the base space and the representation space . . . . . . . . . . . . . . . . 167 54.1.4 Inverse mapping

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

54.1.5 Origin and point at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 54.2 Geometrical objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 54.2.1 As the solution of a pair of equations

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

54.2.2 As derived from points of the object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 54.2.3 odds

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

54.3 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 54.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 54.5 References 54.6 Bibliography

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

54.6.1 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 54.6.2 Online resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 55 Connected ring

173

55.1 Examples and non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 55.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

xii

CONTENTS 55.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

56 Countably generated module

174

56.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 56.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 57 Cyclic module

175

57.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 57.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 57.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 57.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 57.5 References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

58 Dedekind–Hasse norm

177

58.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 58.2 Integral and principal ideal domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 58.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 58.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 58.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 59 Dense submodule

179

59.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 59.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 59.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 59.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 59.4.1 Rational hull of a module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 59.4.2 Maximal right ring of quotients 59.5 References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

60 Depth (ring theory)

182

60.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 60.1.1 Theorem (Rees)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

60.2 Depth and projective dimension 60.3 Depth zero rings 60.4 References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

61 Depth of noncommutative subrings

184

61.1 Definition and first examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 61.2 Depth in relation to Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 61.3 Depth in relation to finite-dimensional semisimple algebras and subgroups of finite groups

. . . . . 185

61.4 Galois theory for depth two extensions and a Main Theorem . . . . . . . . . . . . . . . . . . . . . 185 61.5 References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

62 Derived algebraic geometry

187

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62.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 62.2 References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

62.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 63 Direct sum of modules

188

63.1 Construction for vector spaces and abelian groups

. . . . . . . . . . . . . . . . . . . . . . . . . . 188

63.1.1 Construction for two vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 63.1.2 Construction for two abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 63.2 Construction for an arbitrary family of modules

. . . . . . . . . . . . . . . . . . . . . . . . . . . 189

63.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 63.4 Internal direct sum

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

63.5 Universal property

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

63.6 Grothendieck group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 63.7 Direct sum of modules with additional structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 63.7.1 Direct sum of algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 63.7.2 Direct sum of Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 63.7.3 Direct sum of modules with bilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . . 192 63.7.4 Direct sum of Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 63.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 63.9 References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

64 Divisibility (ring theory)

195

64.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 64.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 64.3 Zero as a divisor, and zero divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 64.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 64.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 64.6 References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

65 Division algebra

197

65.1 Definitions

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

65.2 Associative division algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 65.3 Not necessarily associative division algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 65.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 65.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 65.6 References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

65.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 66 Division ring

200

66.1 Relation to fields and linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 66.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 66.3 Ring theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 66.4 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

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CONTENTS 66.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 66.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 66.7 References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

66.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 66.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 67 Domain (ring theory) 67.1 Examples and non-examples

203 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

67.2 Constructions of domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 67.3 Group rings and the zero divisor problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 67.4 Spectrum of an integral domain

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

67.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 67.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 67.7 References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

68 Double centralizer theorem

206

68.1 Statements of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 68.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 68.1.2 Central simple algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 68.1.3 Artinian rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 68.1.4 Polynomial identity rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 68.2 Double centralizer property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 68.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 68.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 69 Dual module

208

69.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 70 Eilenberg–Mazur swindle

209

70.1 Mazur swindle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 70.2 Eilenberg swindle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 70.3 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 70.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 70.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 70.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 71 Elementary divisors

212

71.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 71.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 72 Endomorphism ring

213

72.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 72.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 72.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

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72.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 72.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 73 Essential extension

216

73.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 73.2 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 73.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 73.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 74 Euclidean domain

218

74.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 74.1.1 Notes on the definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 74.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 74.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 74.4 Norm-Euclidean fields

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

74.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 74.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 74.7 References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

75 Extension of scalars

222

75.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 75.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 75.2.1 Applications

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

75.3 Interpretation as a functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 75.4 Connection with restriction of scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 75.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 75.6 References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

76 Finitely generated module

224

76.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 76.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 76.3 Some facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 76.4 Finitely generated modules over a commutative ring . . . . . . . . . . . . . . . . . . . . . . . . . 225 76.5 Generic rank

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

76.6 Equivalent definitions and finitely cogenerated modules . . . . . . . . . . . . . . . . . . . . . . . . 226 76.7 Finitely presented, finitely related, and coherent modules . . . . . . . . . . . . . . . . . . . . . . . 227 76.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 76.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 76.10Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 77 Finite ring

229

77.1 Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 77.2 Wedderburn’s theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

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CONTENTS 77.3 Finite field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 77.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 77.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

78 Finitely generated module

232

78.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 78.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 78.3 Some facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 78.4 Finitely generated modules over a commutative ring . . . . . . . . . . . . . . . . . . . . . . . . . 233 78.5 Generic rank

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

78.6 Equivalent definitions and finitely cogenerated modules . . . . . . . . . . . . . . . . . . . . . . . . 234 78.7 Finitely presented, finitely related, and coherent modules . . . . . . . . . . . . . . . . . . . . . . . 235 78.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 78.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 78.10Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 79 Fitting lemma

237

79.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 79.2 References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

80 Flat module

238

80.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 80.1.1 Commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 80.1.2 General rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 80.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 80.3 Case of commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 80.4 Categorical colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 80.5 Homological algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 80.6 Flat resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 80.7 In constructive mathematics

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

80.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 80.9 References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

80.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 81 Formal power series

244

81.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 81.2 The ring of formal power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 81.2.1 Definition of the formal power series ring . . . . . . . . . . . . . . . . . . . . . . . . . . 245 81.2.2 Universal property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 81.3 Operations on formal power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 81.3.1 Power series raised to powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 81.3.2 Inverting series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 81.3.3 Dividing series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

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81.3.4 Extracting coefficients

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

81.3.5 Composition of series

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

81.3.6 Composition inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 81.3.7 Formal differentiation of series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 81.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 81.4.1 Algebraic properties of the formal power series ring . . . . . . . . . . . . . . . . . . . . . 251 81.4.2 Topological properties of the formal power series ring . . . . . . . . . . . . . . . . . . . . 252 81.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 81.6 Interpreting formal power series as functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 81.7 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 81.7.1 Formal Laurent series

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

81.7.2 The Lagrange inversion formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 81.7.3 Power series in several variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 81.7.4 Non-commuting variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 81.7.5 On a semiring

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

81.7.6 Replacing the index set by an ordered abelian group . . . . . . . . . . . . . . . . . . . . . 257 81.8 Examples and related topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 81.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 81.10References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

81.11Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 82 Fractional ideal

259

82.1 Definition and basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 82.2 Dedekind domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 82.3 Divisorial ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 82.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 82.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 82.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 83 Free algebra

261

83.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 83.2 Contrast with Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 83.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 83.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 84 Free ideal ring

263

84.1 Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 84.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 84.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 85 Free module

265

85.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 85.2 Formal linear combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

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CONTENTS 85.2.1 Another construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

85.3 Universal property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 85.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 85.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 85.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 85.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 85.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 86 Frobenius algebra

268

86.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 86.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 86.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 86.4 Category-theoretical definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 86.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 86.5.1 Topological quantum field theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 86.6 Generalization: Frobenius extension

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

86.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 86.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 86.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 87 G-domain

274

87.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 88 Gauge theory gravity

275

88.1 Mathematical foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 88.2 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 88.3 Relation to general relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 88.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 88.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 89 GCD domain

278

89.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 89.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 89.3 References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

90 Gelfand ring

280

90.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 91 Generalized Clifford algebra

281

91.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 91.1.1 Abstract definition

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

91.1.2 More specific definition

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

91.2 Matrix representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 91.2.1 Specific examples

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

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91.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 91.4 References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

91.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 92 Geometric algebra

285

92.1 Definition and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 92.1.1 Inner and outer product of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 92.1.2 Blades, grading, and canonical basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 92.1.3 Grade projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 92.1.4 Representation of subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 92.1.5 Unit pseudoscalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 92.1.6 Dual basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 92.1.7 Extensions of the inner and outer products . . . . . . . . . . . . . . . . . . . . . . . . . . 291 92.1.8 Terminology specific to geometric algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 291 92.2 Geometric interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 92.2.1 Projection and rejection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 92.2.2 Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 92.2.3 Hypervolume of an n-parallelotope spanned by n vectors . . . . . . . . . . . . . . . . . . . 295 92.2.4 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 92.3 Linear functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 92.4 Examples and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 92.4.1 Intersection of a line and a plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 92.4.2 Rotating systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 92.4.3 Electrodynamics and special relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 92.5 Relationship with other formalisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 92.6 Geometric calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 92.7 Conformal geometric algebra (CGA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 92.8 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 92.9 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 92.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 92.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 92.12References, and further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 92.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 93 Global dimension

307

93.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 93.2 Alternative characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 93.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 94 Glossary of module theory

309

94.1 Basic definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 94.2 Types of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

xx

CONTENTS 94.3 Operations on modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 94.3.1 Changing scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 94.4 Homological algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 94.5 Modules over special rings

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

94.6 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 94.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 94.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 95 Glossary of ring theory

315

95.1 Definition of a ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 95.2 Types of elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 95.3 Homomorphisms and ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 95.4 Types of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 95.5 Ring constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 95.5.1 Polynomial rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 95.6 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 95.7 Ringlike structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 95.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 95.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 95.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 96 Going up and going down

321

96.1 Going up and going down . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 96.1.1 Lying over and incomparability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 96.1.2 Going-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 96.1.3 Going down

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

96.2 Going-up and going-down theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 96.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 97 Goldie’s theorem

324

97.1 Sketch of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 97.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 97.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 98 Goldman domain

326

98.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 98.2 References 99 Graded ring

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 327

99.1 First properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 99.2 Graded module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 99.3 Invariants of graded modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 99.4 Graded algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

CONTENTS 99.5 G-graded rings and algebras

xxi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

99.6 Anticommutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 99.6.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 99.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 99.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 99.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 100Group algebra

331

100.1Group algebras of topological groups: Cc(G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 100.2The convolution algebra L1 (G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 100.3The group C*-algebra C*(G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 100.3.1 The reduced group C*-algebra Cr*(G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 100.4von Neumann algebras associated to groups

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

100.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 100.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 101Group ring

335

101.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 101.2Two simple examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 101.3Some basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 101.4Group algebra over a finite group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 101.4.1 Interpretation as functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 101.4.2 Regular representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 101.4.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 101.4.4 Representations of a group algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 101.4.5 Center of a group algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 101.5Group rings over an infinite group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 101.6Representations of a group ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 101.7Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 101.7.1 Adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 101.7.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 101.8Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 101.9See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 101.9.1 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 101.9.2 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 101.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 101.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 102Hereditary ring

342

102.1Equivalent definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 102.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 102.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

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102.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 103Hermite ring

344

103.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 104Hilbert–Kunz function 104.1References

345

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

104.2Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 105Hochschild homology

346

105.1Definition of Hochschild homology of algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 105.1.1 Hochschild complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 105.1.2 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 105.2Hochschild homology of functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 105.2.1 Loday functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 105.2.2 Another description of Hochschild homology of algebras . . . . . . . . . . . . . . . . . . 347 105.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 105.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 105.5External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 106Hopfian object

349

106.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 106.2Hopfian and cohopfian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 106.3Hopfian and cohopfian modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 106.4Hopfian and cohopfian rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 106.5Hopfian and cohopfian topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 106.6References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

106.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 107Hopkins–Levitzki theorem

351

107.1Sketch of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 107.2In Grothendieck categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 107.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 107.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 108Ideal (order theory)

353

108.1Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 108.2Prime ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 108.3Maximal ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 108.4Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 108.5History

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

108.6Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 108.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 108.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

CONTENTS

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108.9References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

109Ideal (ring theory) 109.1History

356

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

109.2Definitions

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

109.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 109.4Motivation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

109.5Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 109.6Ideal generated by a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 109.6.1 Example

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

109.7Types of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 109.8Further properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 109.9Ideal operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 109.10Ideals and congruence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 109.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 109.12References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

110Ideal class group

362

110.1History and origin of the ideal class group

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

110.2Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 110.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 110.4Relation with the group of units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 110.5Examples of ideal class groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 110.5.1 Class numbers of quadratic fields 110.6Connections to class field theory

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

110.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 110.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 110.9References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

111Ideal norm

366

111.1Relative norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 111.2Absolute norm

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

111.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 111.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 112Ideal quotient

368

112.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 112.2Calculating the quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 112.3Geometric interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 112.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 113Ideal theory

370

113.1In mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

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113.2In political philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 113.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 114Idealizer

371

114.1Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 114.2Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 114.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 115Indecomposable module

373

115.1Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 115.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 115.2.1 Field

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

115.2.2 PID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 115.3Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 115.4Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 115.5References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

116Injective hull

375

116.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 116.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 116.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 116.3.1 Ring structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 116.4Uniform dimension and injective modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 116.5Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 116.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 116.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 116.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 116.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 117Injective module

378

117.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 117.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 117.2.1 First examples

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

117.2.2 Commutative examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 117.2.3 Artinian examples 117.3Theory

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

117.3.1 Submodules, quotients, products, and sums

. . . . . . . . . . . . . . . . . . . . . . . . . 380

117.3.2 Baer’s criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 117.3.3 Injective cogenerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 117.3.4 Injective hulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 117.3.5 Injective resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 117.3.6 Indecomposables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 117.3.7 Change of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

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117.3.8 Self-injective rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 117.4Generalizations and specializations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 117.4.1 Injective objects

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

117.4.2 Divisible groups

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

117.4.3 Pure injectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 117.5References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

117.5.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 117.5.2 Textbooks

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

117.5.3 Primary sources 118Integer

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 384

118.1Algebraic properties

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

118.2Order-theoretic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 118.3Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 118.4Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 118.5Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 118.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 118.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 118.8References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

118.9Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 118.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 119Integer-valued polynomial

390

119.1Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 119.2Fixed prime divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 119.3Other rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 119.4Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 119.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 119.5.1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 119.5.2 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 119.6Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 120Integral closure of an ideal

392

120.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 120.2Structure results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 120.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 120.4References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

120.5Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 121Integral domain

394

121.1Definitions

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

121.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 121.3Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

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121.4Divisibility, prime elements, and irreducible elements . . . . . . . . . . . . . . . . . . . . . . . . 395 121.5Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 121.6Field of fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 121.7Algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 121.8Characteristic and homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 121.9See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 121.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 121.11References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

122Integral element

399

122.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 122.2Equivalent definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 122.3Integral extensions

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

122.4Integral closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 122.5Conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 122.6Finiteness of integral closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 122.7Noether’s normalization lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 122.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 122.9References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

122.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 123Invariant basis number

405

123.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 123.2Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 123.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 123.4Other results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 123.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 124Invariant factor

407

124.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 124.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 125Irreducible element

408

125.1Relationship with prime elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 125.2Example

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

125.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 125.4References 126Irreducible ideal

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 410

126.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 126.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 127Irreducible ring

411

127.1Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

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127.2Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 127.3Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 128Isotypical representation

413

128.1Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 128.2Example

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

128.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 129Jacobian ideal

414

129.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 130Jacobson density theorem

415

130.1Motivation and formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 130.2Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 130.2.1 Proof of the Jacobson density theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 130.3Topological characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 130.4Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 130.5Relations to other results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 130.6Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 130.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 130.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 131Jacobson radical

418

131.1Intuitive discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 131.2Equivalent characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 131.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 131.4Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 131.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 131.6Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 131.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 132Jacobson ring

422

132.1Jacobson rings and the Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 132.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 132.3Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 132.4Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 132.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 132.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 133Jacobson’s conjecture

424

133.1Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 133.2Partial results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 133.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

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134Jaffard ring

426

134.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 134.2External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 135Kaplansky’s conjecture

427

135.1Kaplansky’s conjectures on groups rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 135.2Kaplansky’s conjecture on Banach algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 135.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 136Kasch ring

429

136.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 136.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 136.3References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430

137Krull ring

431

137.1Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 137.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 137.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 137.4The divisor class group of a Krull ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 137.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 138Krull’s principal ideal theorem

433

138.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 139Krull’s theorem

434

139.1Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 139.2Krull’s Hauptidealsatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 139.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 139.4References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

140Krull–Schmidt theorem

436

140.1Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 140.2Statement of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 140.3Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 140.4Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 140.5Krull–Schmidt Theorem for Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 140.6History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 140.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 140.8References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

140.9Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 140.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 141Kummer ring

439

141.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

CONTENTS

xxix

141.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 142Kurosh problem

440

142.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 143Köthe conjecture

441

143.1Equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 143.2Related problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 143.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 143.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 144Lattice (module)

443

144.1Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 144.2Pure sublattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 144.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 144.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 145Laurent polynomial

444

145.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 145.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 145.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 145.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 146Length of a module

446

146.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 146.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 146.3Facts

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

146.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 146.5References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

147Levitzky’s theorem

448

147.1Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 147.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 147.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 147.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 148Local ring

450

148.1Definition and first consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 148.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 148.2.1 Ring of germs

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

148.2.2 Valuation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 148.2.3 Non-commutative

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

148.3Some facts and definitions

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

148.3.1 Commutative Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

xxx

CONTENTS 148.3.2 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 148.4Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 148.5References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

148.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 149Localization (algebra) 149.1Construction

454

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

149.1.1 Localization of a ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 149.1.2 Localization of a module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 149.2Examples and applications

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

149.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 149.4Stability under localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 149.5Local property

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

149.6Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 149.7(Quasi-)coherent sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 149.8Non-commutative case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 149.9See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 149.9.1 Localization

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

149.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 149.11References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

150Localization of a module

460

150.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 150.2Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 150.3Tensor product interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 150.4Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 150.5(Quasi-)coherent sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 150.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 150.6.1 Localization

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

150.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 151Localization of a ring 151.1Terminology

463

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

151.2Construction and properties for commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . 463 151.2.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 151.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 151.2.3 Properties

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

151.3Category theoretic description

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

151.4Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 151.5Non-commutative case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 151.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 151.6.1 Localization

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

CONTENTS 151.7References

xxxi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

151.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 152Loewy ring

467

152.1Loewy length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 152.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 153Marot ring

468

153.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 154Matrix ring

469

154.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 154.2Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 154.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 154.4Diagonal subring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 154.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 154.5Matrix semiring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 154.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 154.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 155Maximal ideal

472

155.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 155.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 155.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 155.4Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 155.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 156Minimal ideal

475

156.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 156.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 156.3Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 156.4References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

156.5External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 157Minimal prime ideal

477

157.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 157.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 157.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 157.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 158Mitchell’s embedding theorem

479

158.1Sketch of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 158.2References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

xxxii

CONTENTS

159Modular representation theory

481

159.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 159.2Example

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

159.3Ring theory interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 159.4Brauer characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 159.5Reduction (mod p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 159.6Number of simple modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 159.7Blocks and the structure of the group algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 159.8Projective modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 159.9Some orthogonality relations for Brauer characters . . . . . . . . . . . . . . . . . . . . . . . . . . 483 159.10Decomposition matrix and Cartan matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 159.11Defect groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 159.12References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

160Module (mathematics)

486

160.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 160.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 160.1.2 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 160.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 160.3Submodules and homomorphisms

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488

160.4Types of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 160.5Further notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 160.5.1 Relation to representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 160.5.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 160.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 160.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 160.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 160.9External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 161Module of covariants

491

161.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 161.2References 162Monoid ring

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 492

162.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 162.2Universal property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 162.3Augmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 162.4Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 162.5Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 162.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 162.7References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

162.8Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

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xxxiii

163Morita equivalence 163.1Motivation

494

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

163.2Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 163.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 163.4Criteria for equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 163.5Properties preserved by equivalence

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

163.6Further directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 163.7Significance in K-theory

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496

163.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 163.9Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 164Multivector

498

164.1Wedge product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 164.2Area and volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 164.2.1 Multivectors in R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 164.2.2 Multivectors in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 164.3Grassmann coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 164.3.1 Multivectors on P 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 164.3.2 Multivectors on P 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 164.4Clifford product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 164.5Geometric algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 164.5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 164.5.2 Bivectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 164.6Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 164.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 164.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 165Nakayama algebra

506

165.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 166Nakayama’s conjecture

507

166.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 167Necklace ring

508

167.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 167.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 167.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 168Nil ideal

509

168.1Commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 168.2Noncommutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 168.3Relation to nilpotent ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 168.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

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168.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 168.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 169Nilpotent

511

169.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 169.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 169.3Commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 169.4Nilpotent elements in Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 169.5Nilpotency in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 169.6Algebraic nilpotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 169.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 169.8References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

170Nilpotent algebra (ring theory)

514

170.1Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 170.2Nil algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 170.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 170.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 170.5External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 171Nilpotent ideal

516

171.1Relation to nil ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 171.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 171.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 171.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 172Nilradical of a ring

518

172.1Commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 172.2Noncommutative rings

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518

172.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 173Noetherian module

520

173.1Characterizations, properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 173.2Use in other structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 173.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 173.4References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

174Noetherian ring

522

174.1Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 174.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 174.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 174.4Primary decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 174.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 174.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

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174.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 175Noncommutative ring

526

175.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 175.2History

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526

175.3Differences between commutative and noncommutative algebra . . . . . . . . . . . . . . . . . . . 526 175.4Important classes of noncommutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 175.4.1 Division rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 175.4.2 Semisimple rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 175.4.3 Semiprimitive rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 175.4.4 Simple rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 175.5Important theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 175.5.1 Wedderburn’s Little Theorem

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

175.5.2 Artin-Wedderburn Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 175.5.3 Jacobson density theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 175.5.4 Nakayama’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 175.5.5 Noncommutative localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 175.5.6 Morita equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 175.5.7 Brauer group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 175.5.8 Ore conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 175.5.9 Goldie’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 175.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 175.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 175.8Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 176Noncommutative unique factorization domain

532

176.1Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 176.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 176.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 177Novikov ring

533

177.1Novikov numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 177.2Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 177.3References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

177.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534 178Opposite ring

535

178.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 178.2Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 178.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 179Order (ring theory)

536

179.1Algebraic number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536

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179.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 179.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 180Ore algebra

538

180.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 180.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 180.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 181Ore condition

539

181.1General idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 181.2Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 181.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 181.4Multiplicative sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 181.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 181.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 181.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 182Ore extension

542

182.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 182.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 182.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 182.4Elements

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543

182.5Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 182.6References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543

183Outermorphism

544

183.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 183.2Adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 183.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 183.4Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 183.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 183.6External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 184Overring

548

184.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 185Pairing

549

185.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 185.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 185.3Pairings in cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 185.4Slightly different usages of the notion of pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 185.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 185.6External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

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186Paravector

551

186.1Fundamental axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 186.2The Three-dimensional Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 186.2.1 Paravectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 186.2.2 Antiautomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 186.2.3 Grade automorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 186.2.4 Invariant subspaces according to the conjugations . . . . . . . . . . . . . . . . . . . . . . 553 186.2.5 Closed Subspaces respect to the product . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 186.2.6 Scalar Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 186.2.7 Biparavectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 186.2.8 Triparavectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 186.2.9 Pseudoscalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 186.2.10Paragradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 186.2.11Null Paravectors as Projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556 186.2.12Null Basis for the paravector space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 186.3Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 186.4Matrix Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 186.4.1 Conjugations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 186.4.2 Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 186.5Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 186.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 186.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 186.7.1 Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 186.7.2 Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 187Partially ordered ring

560

187.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 187.2f-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 187.2.1 Example 187.2.2 Properties

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

187.3Formally verified results for commutative ordered rings 187.4References

. . . . . . . . . . . . . . . . . . . . . . . 561

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

187.5Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 187.6External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 188Perfect field

563

188.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 188.2Field extension over a perfect field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 188.3Perfect closure and perfection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 188.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 188.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 188.6References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564

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188.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 189Perfect ring

566

189.1Perfect ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 189.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 189.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 189.1.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 189.2Semiperfect ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 189.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 189.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 189.2.3 Properties 189.3References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567

190Plane of rotation

568

190.1Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 190.1.1 Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 190.1.2 Plane of rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 190.2Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 190.3Three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 190.4Four dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 190.4.1 Simple rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 190.4.2 Double rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572 190.4.3 Isoclinic rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572 190.5Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572 190.6Mathematical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572 190.6.1 Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 190.6.2 Bivectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 190.6.3 Eigenvalues and eigenplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 190.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 190.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 190.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 191Plücker coordinates

577

191.1Geometric intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 191.2Algebraic definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 191.2.1 Primal coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 191.2.2 Plücker map

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

191.2.3 Dual coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 191.2.4 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 191.3Bijection between lines and Klein quadric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 191.3.1 Plane equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 191.3.2 Quadratic relation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580

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191.3.3 Point equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 191.3.4 Bijectivity

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581

191.4Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 191.4.1 Line-line crossing

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582

191.4.2 Line-line join . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 191.4.3 Line-line meet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 191.4.4 Plane-line meet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 191.4.5 Point-line join

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582

191.4.6 Line families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 191.4.7 Line geometry

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583

191.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 191.6References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583

192Poisson ring

585

192.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 192.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 193Polynomial identity ring

586

193.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 193.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 193.3PI-rings as generalizations of commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 193.4The set of identities a PI-ring satisfies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 193.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588 193.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588 193.7Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588 193.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588 194Polynomial ring

589

194.1The polynomial ring K[X]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

194.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 194.1.2 Degree of a polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 194.1.3 Properties of K[X] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 194.1.4 Modules

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592

194.2Polynomial evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 194.3The polynomial ring in several variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 194.3.1 Polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592

194.3.2 The polynomial ring

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593

194.3.3 Hilbert’s Nullstellensatz

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593

194.4Properties of the ring extension R ⊂ R[X]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594

194.4.1 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 194.5Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 194.5.1 Infinitely many variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594

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CONTENTS 194.5.2 Generalized exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 194.5.3 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 194.5.4 Noncommutative polynomial rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596 194.5.5 Differential and skew-polynomial rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596 194.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596 194.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596

195Posner’s theorem 195.1References 196Primary ideal

598 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 599

196.1Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 196.2Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 196.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 196.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 197Prime element

601

197.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 197.2Connection with prime ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 197.3Irreducible elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 197.4Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 197.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 198Prime ideal

603

198.1Prime ideals for commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 198.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 198.1.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 198.1.3 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 198.2Prime ideals for noncommutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 198.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 198.3Important facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 198.4Connection to maximality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 198.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 198.6Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 199Prime ring

608

199.1Equivalent definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 199.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 199.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 199.4Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 199.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 200Primitive ideal

610

200.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610

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201Primitive ring

611

201.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 201.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 201.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 201.3.1 Full linear rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612 201.4References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612

202Principal ideal

613

202.1Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 202.2Examples of non-principal ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 202.3Related definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 202.4Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 202.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 202.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 203Principal ideal domain

615

203.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 203.2Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 203.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 203.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 203.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 203.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 203.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 204Principal ideal ring

618

204.1General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 204.2Commutative examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 204.3Structure theory for commutative PIR’s

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619

204.4Noncommutative examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 204.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 205Principal ideal theorem

620

205.1Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620 205.2History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620 205.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620 206Principal indecomposable module

622

206.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 206.2Relations 206.3References 207Product of rings

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 624

207.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624 207.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624

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CONTENTS 207.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 207.4Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 207.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625

208Projective cover

626

208.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 208.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 208.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 208.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 208.5References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627

209Projective line over a ring

628

209.1Instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628 209.2Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 209.2.1 Point-parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 209.3Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 209.4Cross-ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630 209.5History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630 209.6Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630 209.7Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 210Projective module 210.1Definitions

633 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633

210.1.1 Lifting property

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633

210.1.2 Split-exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633 210.1.3 Direct summands of free modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 210.1.4 Exactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 210.1.5 Dual basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 210.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 210.3Projective resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 210.4Projective modules over commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 210.4.1 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 210.5Vector bundles and locally free modules

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636

210.6Projective modules over a polynomial ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 210.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 210.8See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 210.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 211Pseudo-ring

638

211.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 211.2References 212Pseudoscalar

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 639

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212.1Pseudoscalars in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 212.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 212.2Pseudoscalars in geometric algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 213Pure submodule

641

213.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 213.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 213.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 214Quadratic integer

643

214.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 214.2Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 214.3Norm and conjugation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644

214.4Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644 214.5Quadratic integer rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 214.5.1 Examples of complex quadratic integer rings

. . . . . . . . . . . . . . . . . . . . . . . . 645

214.5.2 Examples of real quadratic integer rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 214.5.3 Principal rings of quadratic integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 214.5.4 Euclidean rings of quadratic integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 214.6Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 214.7References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648

215Quadric geometric algebra

649

215.1Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650 216Quasi-Frobenius ring

651

216.1Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 216.2Thrall’s QF-1,2,3 generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 216.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 216.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 216.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 216.6Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 216.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 217Quasiregular element

654

217.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 217.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 217.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 217.4Generalization to semirings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 217.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656 217.6Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656 217.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 217.8See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657

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218Quasisymmetric function 218.1Definitions

658

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658

218.2Important bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658 218.3Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 218.4Related algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 218.5External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660 218.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660 219Quotient module

662

219.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 219.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 219.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 220Quotient ring

663

220.1Formal quotient ring construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 220.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 220.2.1 Alternative complex planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664 220.2.2 Quaternions and alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664 220.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 220.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 220.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 220.6Further references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666 220.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666 221Radical of a module

667

221.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 221.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 221.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 221.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668 222Radical of a ring

669

222.1Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 222.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 222.2.1 The Jacobson radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 222.2.2 The Baer radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 222.2.3 The upper nil radical or Köthe radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 222.2.4 Singular radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 222.2.5 The Levitzki radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 222.2.6 The Brown–McCoy radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 222.2.7 The von Neumann regular radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671 222.2.8 The Artinian radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671 222.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671 222.4References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671

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223Radical of an ideal

672

223.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 223.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 223.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673 223.4Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673 223.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674 223.6Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674 223.7References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674

224Rank ring

675

224.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675 224.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675 225Real closed ring

676

225.1Examples of real closed rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676 225.2Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676 225.3The real closure of a commutative ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677 225.4Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677 225.5Model theoretic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678 225.6Comparison with characterizations of real closed fields . . . . . . . . . . . . . . . . . . . . . . . . 678 225.7References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678

226Reduced ring

679

226.1Examples and non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 226.2Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 226.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 226.4Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680 226.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680 227Torsionless module 227.1Properties and examples

681 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681

227.2Relation with semihereditary rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 227.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 227.4References 228Regev’s theorem

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 683

228.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683 229Regular ideal

684

229.1Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684 229.1.1 Modular ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684 229.1.2 Regular element ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684 229.1.3 Von Neumann regular ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 229.1.4 Quotient von Neumann regular ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685

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CONTENTS 229.2References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686

230Regular local ring

687

230.1Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687 230.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688 230.3Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688 230.4Origin of basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688 230.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 230.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 231Regular ring

690

231.1Noncommutative ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690 231.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690 231.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690 232Remak decomposition

691

232.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691 233Resolution (algebra)

692

233.1Resolutions of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692 233.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692 233.1.2 Free, projective, injective, and flat resolutions . . . . . . . . . . . . . . . . . . . . . . . . 693 233.1.3 Graded modules and algebras

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693

233.1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693 233.2Resolutions in abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693 233.3Acyclic resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694 233.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694 233.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694 233.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 234Riemann–Silberstein vector

696

234.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696 234.2Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696 234.3Photon wave function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 234.3.1 Schrödinger equation for the photon and the Heisenberg uncertainty relations . . . . . . . . 697 234.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 234.5Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 235Ring (mathematics)

700

235.1Definition and illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700 235.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700 235.1.2 Notes on the definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702 235.1.3 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702 235.1.4 Example: Integers modulo 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702

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235.1.5 Example: 2-by-2 matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 235.2History

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703

235.2.1 Dedekind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 235.2.2 Hilbert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 235.2.3 Fraenkel and Noether . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 235.2.4 Multiplicative identity: mandatory or optional? . . . . . . . . . . . . . . . . . . . . . . . . 703 235.3Basic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 235.4Basic concepts

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706

235.4.1 Elements in a ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 235.4.2 Subring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 235.4.3 Ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 235.4.4 Homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 235.4.5 Quotient ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 235.5Ring action: a module over a ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 235.6Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709 235.6.1 Direct product

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709

235.6.2 Polynomial ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710 235.6.3 Matrix ring and endomorphism ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 235.6.4 Limits and colimits of rings 235.6.5 Localization

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712

235.6.6 Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713 235.6.7 Rings with generators and relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713 235.7Special kinds of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713 235.7.1 Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713 235.7.2 Division ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 235.7.3 Semisimple rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 235.7.4 Central simple algebra and Brauer group . . . . . . . . . . . . . . . . . . . . . . . . . . . 715 235.7.5 Valuation ring

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715

235.8Rings with extra structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 235.9Some examples of the ubiquity of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 235.9.1 Cohomology ring of a topological space . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 235.9.2 Burnside ring of a group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 235.9.3 Representation ring of a group ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 235.9.4 Function field of an irreducible algebraic variety . . . . . . . . . . . . . . . . . . . . . . . 717 235.9.5 Face ring of a simplicial complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717 235.10Category theoretical description 235.11Generalization

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717

235.11.1Rng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717 235.11.2Nonassociative ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718 235.11.3Semiring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718 235.12Other ring-like objects

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718

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CONTENTS 235.12.1Ring object in a category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718 235.12.2Ring scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718 235.12.3Ring spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718

235.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718 235.14Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 235.15Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720 235.16References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721

235.16.1General references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721 235.16.2Special references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722 235.16.3Primary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723 235.16.4Historical references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723 236Ring homomorphism

724

236.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724 236.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725 236.3The category of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725 236.3.1 Endomorphisms, isomorphisms, and automorphisms . . . . . . . . . . . . . . . . . . . . . 726 236.3.2 Monomorphisms and epimorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726 236.4Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726 236.5References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726

236.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726 237Ring of integers

727

237.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727 237.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727 237.3Multiplicative structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727 237.4Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 237.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 237.6Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 238Ring of polynomial functions 238.1Symmetric multilinear maps

729 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729

238.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730 238.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730 238.4References 239Ring theory

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730 731

239.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 239.2Commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732 239.2.1 Algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732 239.3Noncommutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732 239.3.1 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732 239.4Some useful theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733

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239.5Structures and invariants of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 239.5.1 Dimension of a commutative ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 239.5.2 Morita equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734 239.5.3 Finitely generated projective module over a ring and Picard group . . . . . . . . . . . . . . 734 239.5.4 Structure of noncommutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734 239.6Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735 239.6.1 The ring of integers of a number field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735 239.6.2 The coordinate ring of an algebraic variety . . . . . . . . . . . . . . . . . . . . . . . . . . 735 239.6.3 Ring of invariants

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735

239.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735 239.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736 240Rng (algebra)

737

240.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737 240.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737 240.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738 240.4Adjoining an identity element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738 240.5Properties weaker than having an identity

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739

240.6Rng of square zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739 240.7Unital homomorphism

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739

240.8See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740 240.9Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740 240.10References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740

241Rotor (mathematics)

741

241.1Generation using reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741 241.1.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741 241.1.2 Restricted alternative formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742 241.1.3 Rotations of multivectors and spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742 241.2Homogeneous representation algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743 241.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743 242SBI ring

744

242.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744 242.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744 243Schanuel’s lemma

745

243.1Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 243.2Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 243.3Long exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746 243.4Origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746 243.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746

l

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244Schreier domain

747

244.1References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747

245Semi-local ring

748

245.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748 245.2Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748 246Semi-orthogonal matrix

749

246.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749 247Semi-simplicity

750

247.1Introductory example of vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750 247.2Semi-simple modules and rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750 247.3Semi-simple matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751 247.4Semi-simple categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751 247.5Semi-simplicity in representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751 247.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752 247.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752 247.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752 248Seminormal ring

753

248.1References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753

249Semiprime ring

754

249.1Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754 249.2General properties of semiprime ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755 249.3Semiprime Goldie rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755 249.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755 249.5External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756 250Semiprimitive ring

757

250.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757 250.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757 250.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757 251Semiring

759

251.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759 251.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760 251.2.1 In general . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760 251.2.2 Specific examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760 251.3Semiring theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761 251.4Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761 251.5Complete and continuous semirings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761 251.6Star semirings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762

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li

251.6.1 Complete star semirings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762 251.7Further generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763 251.8Semiring of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763 251.9Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763 251.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763 251.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763 251.12Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763 251.13Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765 252Semisimple module

766

252.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766 252.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766 252.3Endomorphism rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767 252.4Semisimple rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767 252.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767 252.4.2 Simple rings

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767

252.4.3 Jacobson semisimple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768 252.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768 252.6References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768

252.6.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768 252.6.2 Textbooks

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768

253Serial module

769

253.1Properties of uniserial and serial rings and modules . . . . . . . . . . . . . . . . . . . . . . . . . . 769 253.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770 253.3Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770 253.4A decomposition uniqueness property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770 253.5Notes on alternate, similar and related terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771 253.6Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771 253.7Primary Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772 254Serre’s criterion on normality

773

254.1Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773 254.1.1 Sufficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773 254.1.2 Necessity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774 254.2Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774 254.3References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774

255Severi–Brauer variety

775

255.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776 255.2Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776 255.3External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776

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256Simple algebra

777

256.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 256.2Simple universal algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 256.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 256.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778 257Simple module

779

257.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779 257.2Basic properties of simple modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779 257.3Simple modules and composition series 257.4The Jacobson density theorem

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 780

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 780

257.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781 257.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781 258Simple ring

782

258.1Wedderburn’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782 258.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783 258.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783 259Simplicial commutative ring

784

259.1Graded ring structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784 259.2Spec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784 259.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784 259.4References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784

260Singular submodule

786

260.1Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786 260.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786 260.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787 260.4Important theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787 260.5Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787 260.6Primary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787 261Skolem–Noether theorem

788

261.1Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788 261.2Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788 261.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789 261.4References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789

262Socle (mathematics)

790

262.1Socle of a group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790 262.2Socle of a module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790 262.3Socle of a Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791 262.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791

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262.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791 263Spacetime algebra

792

263.1Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792 263.2Reciprocal frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792 263.3Spacetime gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793 263.4Spacetime split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793 263.5Multivector division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793 263.6Spacetime algebra description of non-relativistic physics . . . . . . . . . . . . . . . . . . . . . . . 794 263.6.1 Non-relativistic quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794 263.7Spacetime algebra description of relativistic physics . . . . . . . . . . . . . . . . . . . . . . . . . 794 263.7.1 Relativistic quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794 263.7.2 A new formulation of General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . 795 263.8See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795 263.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795 263.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796 264Spectrum of a ring

797

264.1Zariski topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797 264.2Sheaves and schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797 264.3Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798 264.4Motivation from algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798 264.5Global Spec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798 264.6Representation theory perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798 264.7Functional analysis perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799 264.8Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 800 264.9See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 800 264.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 800 264.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 800 265Square-free

801

265.1Alternate characterizaions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801 265.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801 265.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801 265.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801 266Stably finite ring

802

266.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802 267Stably free module

803

267.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803 267.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803 267.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803

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CONTENTS 267.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803

268Structure theorem for finitely generated modules over a principal ideal domain

804

268.1Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804 268.1.1 Invariant factor decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804 268.1.2 Primary decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 268.2Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 268.3Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806 268.4Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806 268.5Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806 268.5.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806 268.5.2 Primary decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806 268.5.3 Indecomposable modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807 268.5.4 Non-finitely generated modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807 268.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807 269Subring

808

269.1Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808 269.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808 269.3Subring test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808 269.4Ring extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808 269.5Subring generated by a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809 269.6Relation to ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809 269.7Profile by commutative subrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809 269.8See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809 269.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809 270Supermodule

810

270.1Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810 270.2Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810 270.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811 271Support of a module

812

271.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812 271.2References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812

272Sylvester domain

813

272.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813 273Symmetric algebra

814

273.1Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814 273.1.1 Grading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814 273.1.2 Distinction with symmetric tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815 273.2Interpretation as polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815

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273.2.1 Symmetric algebra of an affine space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816 273.3Categorical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816 273.4Analogy with exterior algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816 273.5Module analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816 273.6As a universal enveloping algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816 273.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816 273.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817 274Syzygy (mathematics)

818

274.1Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818 275Tensor product of algebras

819

275.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819 275.2Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 820 275.3References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 820

276Tensor product of modules

821

276.1Balanced product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821 276.2Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821 276.2.1 Tensor product of linear maps and a change of base ring

. . . . . . . . . . . . . . . . . . 824

276.2.2 Several modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824 276.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824 276.3.1 Extension of scalars

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827

276.4Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828 276.5Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829 276.6As linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830 276.6.1 Dual module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830 276.6.2 Duality pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830 276.6.3 An element as a (bi)linear map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831 276.6.4 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831 276.7Example from differential geometry: tensor field . . . . . . . . . . . . . . . . . . . . . . . . . . . 831 276.8Relationship to flat modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832 276.9Additional structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832 276.10Generalization

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833

276.10.1Tensor product of complexes of modules

. . . . . . . . . . . . . . . . . . . . . . . . . . 833

276.10.2Tensor product of sheaves of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833 276.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833 276.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833 276.13References 277Tight closure

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834 835

277.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835

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278Top (mathematics)

836

278.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836 278.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836 279Topological ring

837

279.1General comments

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837

279.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837 279.3Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837 279.4Topological fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 838 279.5References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 838

280Torsion (algebra)

839

280.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839 280.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839 280.3Case of a principal ideal domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 840 280.4Torsion and localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 840 280.5Torsion in homological algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841 280.6Abelian varieties

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841

280.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841 280.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841 281Torsion-free module

843

281.1Examples of torsion-free modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843 281.2Structure of torsion-free modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843 281.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843 281.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844 282Torsionless module

845

282.1Properties and examples

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845

282.2Relation with semihereditary rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846 282.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846 282.4References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846

283Total ring of fractions

847

283.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847 283.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847 283.3The total ring of fractions of a reduced ring 283.4Generalization

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 848

283.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 848 283.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 848 284Triangular matrix ring

849

284.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849 284.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849

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lvii

285Uniform module

850

285.1Properties and examples of uniform modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 850 285.2Uniform dimension of a module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 850 285.3Hollow modules and co-uniform dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 851 285.4Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 851 285.5Primary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 851 286Unipotent

853

286.1Unipotent algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853 286.2Unipotent radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853 286.3Jordan decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854 286.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854 286.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854 287Unique factorization domain

855

287.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855 287.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855 287.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856 287.4Equivalent conditions for a ring to be a UFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857 287.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857 287.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857 288Unit (ring theory)

859

288.1Group of units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859 288.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 860 288.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 860 289Universal enveloping algebra

861

289.1Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 861 289.2Universal property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 861 289.3Direct construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862 289.4Examples in particular cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862 289.5Further description of structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862 289.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863 289.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863 290Universal geometric algebra

864

290.1Vector manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864 290.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865 290.3References 291Valuation ring

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865 866

291.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866 291.2Definitions

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867

lviii

CONTENTS 291.3Construction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867

291.4Dominance and integral closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867 291.5Ideals in valuation rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 868 291.6Places . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869 291.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869 291.8References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 870

292Von Neumann regular ring

871

292.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871 292.2Facts

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871

292.3Generalizations and specializations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872 292.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873 292.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873 292.6Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873 293Weak dimension

874

293.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874 293.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874 294Wedderburn’s little theorem 294.1History

875

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875

294.2Relationship to the Brauer group of a finite field . . . . . . . . . . . . . . . . . . . . . . . . . . . 875 294.3Sketch of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875 294.4Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876 294.5References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877

294.6External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877 295Weyl algebra 295.1Generators and relations

878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 878

295.1.1 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 878 295.2Properties of the Weyl algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879 295.2.1 Positive characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879 295.3Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879 295.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 880 296Witt vector

881

296.1Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 881 296.1.1 Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 881 296.2Construction of Witt rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882 296.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883 296.4Universal Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883 296.5Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883 296.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884

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296.5.2 Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884 296.5.3 Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884 296.6Ring schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885 296.7Commutative unipotent algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885 296.8See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885 296.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885 297Zero divisor

886

297.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886 297.1.1 One-sided zero-divisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886 297.2Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887 297.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887 297.4Zero as a zero divisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887 297.5Zero divisor on a module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887 297.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 888 297.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 888 297.8References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 888

298Zero object (algebra)

889

298.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 890 298.1.1 Unital structures

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 890

298.2Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 891 298.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 891 298.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 891 299Zero ring

892

299.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892 299.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892 299.3Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893 299.4Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893 299.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894 300Zero-product property

895

300.1Algebraic context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895 300.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895 300.3Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896 300.4Application to finding roots of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896 300.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896 300.6Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897 300.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897 300.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897 301Zorn ring

898

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CONTENTS 301.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898

302λ-ring

899

302.1Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 899 302.2Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 900 302.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 900 302.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 901 302.5Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 901 302.6Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 902 302.6.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 902 302.6.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919 302.6.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922

Chapter 1

*-algebra In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings R and A, where R is commutative and A has the structure of an associative algebra over R. Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints.

1.1 Terminology 1.1.1

*-ring

In mathematics, a *-ring is a ring with a map * : A → A that is an antiautomorphism and an involution. More precisely, * is required to satisfy the following properties:[1] • (x + y)* = x* + y* • (x y)* = y* x* • 1* = 1 • (x*)* = x for all x, y in A. This is also called an involutive ring, involutory ring, and ring with involution. Note that the third axiom is actually redundant, because the second and fourth axioms imply 1* is also a multiplicative identity, and identities are unique. Elements such that x* = x are called self-adjoint.[2] Archetypical examples of a *-ring are fields of complex numbers and algebraic numbers with complex conjugation as the involution. One can define a sesquilinear form over any *-ring. Also, one can define *-versions of algebraic objects, such as ideal and subring, with the requirement to be *-invariant: x ∈ I ⇒ x* ∈ I and so on.

1.1.2

*-algebra

A *-algebra A is a *-ring,[3] with involution * that is an associative algebra over a commutative *-ring R with involution ′, such that (r x)* = r′ x* ∀r ∈ R, x ∈ A.[4] The base *-ring is usually the complex numbers (with ′ acting as complex conjugation) and is commutative with A such that A is both left and right algebra. Since R is central in A, that is, 1

2

CHAPTER 1. *-ALGEBRA rx = xr ∀r ∈ R, x ∈ A

the * on A is conjugate-linear in R, meaning (λ x + μ y)* = λ′ x* + μ′ y* for λ, μ ∈ R, x, y ∈ A. A *-homomorphism f : A → B is an algebra homomorphism that is compatible with the involutions of A and B, i.e., • f(a*) = f(a)* for all a in A.[2]

1.1.3

*-operation

A *-operation on a *-ring is an operation on a ring that behaves similarly to complex conjugation on the complex numbers. A *-operation on a *-algebra is an operation on an algebra over a *-ring that behaves similarly to taking adjoints in GLn(C).

1.1.4

Notation

The * involution is a unary operation written with a postfixed star glyph centered above or near the mean line: x ↦ x*, or x ↦ x∗ (TeX: x^*), but not as "x∗"; see the asterisk article for details.

1.2 Examples • Any commutative ring becomes a *-ring with the trivial (identical) involution. • The most familiar example of a *-ring and a *-algebra over reals is the field of complex numbers C where * is just complex conjugation. • More generally, a field extension made by adjunction of a square root (such as the imaginary unit √−1) is a *-algebra over the original field, considered as a trivially-*-ring. The * flips the sign of that square root. • A quadratic integer ring (for some D) is a commutative *-ring with the * defined in the similar way; quadratic fields are *-algebras over appropriate quadratic integer rings. • Quaternions, split-complex numbers, dual numbers, and possibly other hypercomplex number systems form *rings (with their built-in conjugation operation) and *-algebras over reals (where * is trivial). Note that neither of the three is a complex algebra. • Hurwitz quaternions form a non-commutative *-ring with the quaternion conjugation. • The matrix algebra of n × n matrices over R with * given by the transposition. • The matrix algebra of n × n matrices over C with * given by the conjugate transpose. • Its generalization, the Hermitian adjoint in the algebra of bounded linear operators on a Hilbert space also defines a *-algebra. • The polynomial ring R[x] over a commutative trivially-*-ring R is a *-algebra over R with P *(x) = P (−x). • If (A, +, ×, *) is simultaneously a *-ring, an algebra over a ring R (commutative), and (r x)* = r (x*) ∀r ∈ R, x ∈ A, then A is a *-algebra over R (where * is trivial).

1.3. ADDITIONAL STRUCTURES

3

• As a partial case, any *-ring is a *-algebra over integers. • Any commutative *-ring is a *-algebra over itself and, more generally, over any its *-subring. • For a commutative *-ring R, its quotient by any its *-ideal is a *-algebra over R. • For example, any commutative trivially-*-ring is a *-algebra over its dual numbers ring, a *-ring with non-trivial *, because the quotient by ε = 0 makes the original ring. • The same about a commutative ring K and its polynomial ring K[x]: the quotient by x = 0 restores K. • In Hecke algebra, an involution is important to the Kazhdan–Lusztig polynomial. • The endomorphism ring of an elliptic curve becomes a *-algebra over the integers, where the involution is given by taking the dual isogeny. A similar construction works for abelian varieties with a polarization, in which case it is called the Rosati involution (see Milne’s lecture notes on abelian varieties). Involutive Hopf algebras are important examples of *-algebras (with the additional structure of a compatible comultiplication); the most familiar example being: • The group Hopf algebra: a group ring, with involution given by g ↦ g−1 .

1.3 Additional structures Many properties of the transpose hold for general *-algebras: • The Hermitian elements form a Jordan algebra; • The skew Hermitian elements form a Lie algebra; • If 2 is invertible in the *-ring, then 1/2(1 + *) and 1/2(1 − *) are orthogonal idempotents,[2] called symmetrizing and anti-symmetrizing, so the algebra decomposes as a direct sum of modules (vector spaces if the *-ring is a field) of symmetric and anti-symmetric (Hermitian and skew Hermitian) elements. These spaces do not, generally, form associative algebras, because the idempotents are operators, not elements of the algebra.

1.3.1

Skew structures

Given a *-ring, there is also the map −* : x ↦ −x*. It does not define a *-ring structure (unless the characteristic is 2, in which case −* is identical to the original *), as 1 ↦ −1, neither is it antimultiplicative, but it satisfies the other axioms (linear, involution) and hence is quite similar to *-algebra where x ↦ x*. Elements fixed by this map (i.e., such that a = −a*) are called skew Hermitian. For the complex numbers with complex conjugation, the real numbers are the Hermitian elements, and the imaginary numbers are the skew Hermitian.

1.4 See also • Semigroup with involution • B*-algebra • C*-algebra • Dagger category • von Neumann algebra • Baer ring • operator algebra • conjugate (algebra) • Cayley–Dickson construction

4

CHAPTER 1. *-ALGEBRA

1.5 Notes and references [1] Weisstein, Eric (2015). “C-Star Algebra”. http://mathworld.wolfram.com/C-Star-Algebra.html''. Weisstein, Eric W. [2] “Octonions”. 2015. Archived from the original on 2015-03-25. Retrieved 2015. [3] Most definitions do not require a *-algebra to have the unity, i.e. a *-algebra is allowed to be a *-rng only. [4] star-algebra in nLab

Chapter 2

Acceptable ring In mathematics, an acceptable ring is a generalization of an excellent ring, with the conditions about regular rings in the definition of an excellent ring replaced by conditions about Gorenstein rings. Acceptable rings were introduced by Sharp (1977). All finite-dimensional Gorenstein rings are acceptable, as are all finitely generated algebras over acceptable rings and all localizations of acceptable rings.

2.1 References • Sharp, Rodney Y. (1977), “Acceptable rings and homomorphic images of Gorenstein rings”, Journal of Algebra 44: 246–261, doi:10.1016/0021-8693(77)90180-6, MR 0441957

5

Chapter 3

Additive identity In mathematics the additive identity of a set which is equipped with the operation of addition is an element which, when added to any element x in the set, yields x. One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings.

3.1 Elementary examples • The additive identity familiar from elementary mathematics is zero, denoted 0. For example, 5+0=5=0+5 • In the natural numbers N and all of its supersets (the integers Z, the rational numbers Q, the real numbers R, or the complex numbers C), the additive identity is 0. Thus for any one of these numbers n, n+0=n=0+n

3.2 Formal definition Let N be a set which is closed under the operation of addition, denoted +. An additive identity for N is any element e such that for any element n in N, e+n=n=n+e Example: The formula is n + 0 = n = 0 + n.

3.3 Further examples • In a group the additive identity is the identity element of the group, is often denoted 0, and is unique (see below for proof). • A ring or field is a group under the operation of addition and thus these also have a unique additive identity 0. This is defined to be different from the multiplicative identity 1 if the ring (or field) has more than one element. If the additive identity and the multiplicative identity are the same, then the ring is trivial (proved below). • In the ring Mm×n(R) of m by n matrices over a ring R, the additive identity is denoted 0 and is the m by n matrix whose entries consist entirely of the identity element 0 in R. For example, in the 2 by 2 matrices over the integers M2 (Z) the additive identity is ( ) 0 0 0= 0 0 6

3.4. PROOFS

7

• In the quaternions, 0 is the additive identity. • In the ring of functions from R to R, the function mapping every number to 0 is the additive identity. • In the additive group of vectors in Rn , the origin or zero vector is the additive identity.

3.4 Proofs 3.4.1

The additive identity is unique in a group

Let (G, +) be a group and let 0 and 0' in G both denote additive identities, so for any g in G, 0 + g = g = g + 0 and 0' + g = g = g + 0' It follows from the above that (0') = (0') + 0 = 0' + (0) = (0)

3.4.2

The additive identity annihilates ring elements

In a system with a multiplication operation that distributes over addition, the additive identity is a multiplicative absorbing element, meaning that for any s in S, s·0 = 0. This can be seen because:

s · 0 = s · (0 + 0) = s · 0 + s · 0 ⇒s·0=s·0−s·0 ⇒s·0=0

3.4.3

The additive and multiplicative identities are different in a non-trivial ring

Let R be a ring and suppose that the additive identity 0 and the multiplicative identity 1 are equal, or 0 = 1. Let r be any element of R. Then r=r×1=r×0=0 proving that R is trivial, that is, R = {0}. The contrapositive, that if R is non-trivial then 0 is not equal to 1, is therefore shown.

3.5 See also • 0 (number) • Additive inverse • Identity element • Multiplicative identity

3.6 References • David S. Dummit, Richard M. Foote, Abstract Algebra, Wiley (3d ed.): 2003, ISBN 0-471-43334-9.

8

CHAPTER 3. ADDITIVE IDENTITY

3.7 External links • uniqueness of additive identity in a ring at PlanetMath.org. • Margherita Barile, “Additive Identity”, MathWorld.

Chapter 4

Additive map For additive functions in number theory, see Additive function. In algebra an additive map, Z-linear map or additive function is a function that preserves the addition operation:

f (x + y) = f (x) + f (y). for any two elements x and y in the domain. For example, any linear map is additive. When the domain is the real numbers, this is Cauchy’s functional equation. For a specific case of this definition, see additive polynomial. Any homomorphism f between abelian groups is additive by this definition. More formally, an additive map of ring R1 into ring R2 is a homomorphism

f : R1 → R2 , of the additive group of R1 into the additive group of R2 . An additive map is not required to preserve the product operation of the ring. If f and g are additive maps, then the map f + g (defined pointwise) is additive.

4.1 Additive map of a division ring Let D be a division ring of characteristic 0 . We can represent an additive map f : D → D of the division ring D as

f (x) = (s)0 f x (s)1 f . We assume a sum over the index s . The number of items depends on the function f . The expressions (s)0 f, (s)1 f ∈ D are called the components of the additive map.

4.2 References • Leslie Hogben, Richard A. Brualdi, Anne Greenbaum, Roy Mathias, Handbook of linear algebra, CRC Press, 2007 • Roger C. Lyndon, Paul E. Schupp, Combinatorial Group Theory, Springer, 2001

9

Chapter 5

Algebra (ring theory) In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R. In this article, all rings are assumed to be unital.

5.1 Formal definition Let R be a commutative ring. An R-algebra is an R-module A together with a binary operation [·, ·]

[·, ·] : A × A → A called A-multiplication, which satisfies the following axiom: • Bilinearity:

[αx + βy, z] = α[x, z] + β[y, z],

[z, αx + βy] = α[z, x] + β[z, y]

for all scalars α , β in R and all elements x, y, z in A.

5.2 Example 5.2.1

Split-biquaternions

Main article: Split-biquaternion The split-biquaternions are an example of an algebra over a ring that is not a field. The base ring of the split-biquaternions is the ring of split-complex numbers (or hyperbolic numbers, also perplex numbers), which are a two-dimensional commutative algebra over the real numbers different from the complex numbers. Every split-complex number has the form x + y j, where x and y are real numbers. The number j is similar to the imaginary unit i, except that j 2 = +1. 10

5.3. ASSOCIATIVE ALGEBRAS

11

A split-biquaternion is a hypercomplex number of the form

q = w + xi + yj + zk where w, x, y, and z are split-complex numbers and i, j, and k multiply as in the quaternion group. Since each coefficient w, x, y, z spans two real dimensions, the split-biquaternion is an element of an eight-dimensional vector space. Considering that it carries a multiplication, this vector space is an algebra over the real field, or an algebra over a ring where the split-complex numbers form the ring. This algebra was introduced by William Kingdon Clifford in an 1873 article for the London Mathematical Society. It has been repeatedly noted in mathematical literature since then, variously as a deviation in terminology, an illustration of the tensor product of algebras, and as an illustration of the direct sum of algebras.

5.3 Associative algebras Main article: Associative algebra If A is a monoid under A-multiplication (it satisfies associativity and it has an identity), then the R-algebra is called an associative algebra. An associative algebra forms a ring over R and provides a generalization of a ring. An equivalent definition of an associative R-algebra is a ring homomorphism f : R → A such that the image of f is contained in the center of A. If the ring B is a commutative ring, a simpler, alternative definition is: Given a ring homomorphism λ : A → B we say that B is an A-algebra.[1] A ring homomorphism ρ : A → B shall always map the identity of A to the identity of B. We also say that B/A is an algebra over A given by ρ . Every ring is a Z -algebra.[2]

5.4 Non-associative algebras Main article: Non-associative algebra A non-associative algebra[3] (or distributive algebra) over a field (or a commutative ring) K is a K-vector space (or more generally a module[4] ) A equipped with a K-bilinear map A × A → A which establishes a binary multiplication operation on A. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (ab)(cd), (a(bc))d and a(b(cd)) may all yield different answers.

5.5 See also • Abelian algebra • Algebraic structure (a much more general term) • Associative algebra • Coalgebra • Graded algebra • Lie algebra • Semiring • Split-biquaternion (example) • Example of a non-associative algebra (example)

12

CHAPTER 5. ALGEBRA (RING THEORY)

5.6 References [1] H. Matsumura Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8. [2] Ernst Kunz, “Introduction to Commutative algebra and algebraic geometry”, Birkhauser 1985, ISBN 0-8176-3065-1 [3] Schafer 1966, Chapter 1. [4] Schafer 1966, pp.1.

5.7 Further reading • Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: SpringerVerlag, ISBN 978-0-387-95385-4, MR 1878556 • Schafer, Richard D. (1995) [1966], An Introduction to Nonassociative Algebras, Dover, ISBN 0-486-68813-5, Zbl 0145.25601

Chapter 6

Algebra homomorphism A homomorphism between two algebras, A and B, over a field (or ring) K, is a map F : A → B such that for all k in K and x,y in A, • F(kx) = kF(x) • F(x + y) = F(x) + F(y) • F(xy) = F(x)F(y)[1][2] If F is bijective then F is said to be an isomorphism between A and B. A common abbreviation for “homomorphism between algebras” is “algebra homomorphism” or “algebra map”. Every algebra homomorphism is a homomorphism of K-modules

6.1 Unital algebra homomorphisms If A and B are two unital algebras, then an algebra homomorphism F : A → B is said to be unital if it maps the unity of A to the unity of B. Often the words “algebra homomorphism” are actually used in the meaning of “unital algebra homomorphism”, so non-unital algebra homomorphisms are excluded. A unital algebra homomorphism is a ring homomorphism.

6.2 Examples Let A = K[x] be the set of all polynomials over a field K and B be the set of all polynomial functions over K. Both A and B are algebras over K given by the standard multiplication and addition of polynomials and functions, respectively. We can map each f in A to fˆ in B by the rule fˆ(t) = f (t) . A routine check shows that the mapping f 7→ fˆ is a homomorphism of the algebras A and B. This homomorphism is an isomorphism if and only if K is an infinite field. Proof. If K is a finite field then let

p(x) =

∏

(x − t).

t∈K

p is a nonzero polynomial in K[x], however p(t) = 0 for all t in K, so pˆ = 0 is the zero function and our homomorphism is not an isomorphism (and, actually, the algebras are not isomorphic, since the algebra of polynomials is infinite while that of polynomial functions is finite). If K is infinite then choose a polynomial f such that fˆ = 0 . We want to show this implies that f = 0 . Let deg f = n and let t0 , t1 , . . . , tn be n + 1 distinct elements of K. Then f (ti ) = 0 for 0 ≤ i ≤ n and by Lagrange 13

14

CHAPTER 6. ALGEBRA HOMOMORPHISM

interpolation we have f = 0 . Hence the mapping f 7→ fˆ is injective. Since this mapping is clearly surjective, it is bijective and thus an algebra isomorphism of A and B. If A is a subalgebra of B, then for every invertible b in B the function that takes every a in A to b−1 a b is an algebra homomorphism (in case A = B , this is called an inner automorphism of B). If A is also simple and B is a central simple algebra, then every homomorphism from A to B is given in this way by some b in B; this is the Skolem-Noether theorem.

6.3 See also • Augmentation (algebra)

6.4 References [1] Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9. [2] Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.

Chapter 7

Algebra of physical space In physics, the algebra of physical space (APS) is the use of the Clifford or geometric algebra Cℓ3 of the threedimensional Euclidean space as a model for (3+1)-dimensional space-time, representing a point in space-time via a paravector (3-dimensional vector plus a 1-dimensional scalar). The Clifford algebra Cℓ3 has a faithful representation, generated by Pauli matrices, on the spin representation C2 ; further, Cℓ3 is isomorphic to the even subalgebra of the 3+1 Clifford algebra, Cℓ0 3,1. APS can be used to construct a compact, unified and geometrical formalism for both classical and quantum mechanics. APS should not be confused with spacetime algebra (STA), which concerns the Clifford algebra Cℓ₁,₃(R) of the four-dimensional Minkowski spacetime.

7.1 Special relativity 7.1.1

Space-time position paravector

In APS, the space-time position is represented as a paravector

x = x0 + x1 e1 + x2 e2 + x3 e3 , where the time is given by the scalar part x0 = t, and e1 , e2 , e3 are the standard basis for position space. Throughout, units such that c = 1 are used, called natural units. In the Pauli matrix representation, the unit basis vectors are replaced by the Pauli matrices and the scalar part by the identity matrix. This means that the Pauli matrix representation of the space-time position is ( x→

x0 + x3 x1 + ix2

7.1.2

x1 − ix2 x0 − x3

)

Lorentz transformations and rotors

Main articles: Lorentz transformation and Rotor (mathematics) The restricted Lorentz transformations that preserve the direction of time and include rotations and boosts can be performed by an exponentiation of the space-time rotation biparavector W

1

L = e2W 15

16

CHAPTER 7. ALGEBRA OF PHYSICAL SPACE

In the matrix representation the Lorentz rotor is seen to form an instance of the SL(2,C) group (special linear group of degree 2 over the complex numbers), which is the double cover of the Lorentz group. The unimodularity of the Lorentz rotor is translated in the following condition in terms of the product of the Lorentz rotor with its Clifford conjugation

¯ = LL ¯ =1 LL This Lorentz rotor can be always decomposed in two factors, one Hermitian B = B† , and the other unitary R† = R−1 , such that

L = BR The unitary element R is called a rotor because this encodes rotations, and the Hermitian element B encodes boosts.

7.1.3

Four-velocity paravector

The four-velocity also called proper velocity is defined as the derivative of the space-time position paravector with respect to proper time τ:

u=

[ ] dx dx0 d 1 dx0 d = + (x e1 + x2 e2 + x3 e3 ) = 1 + 0 (x1 e1 + x2 e2 + x3 e3 ) . dτ dτ dτ dτ dx

This expression can be brought to a more compact form by defining the ordinary velocity as

v=

d (x1 e1 + x2 e2 + x3 e3 ) dx0

and recalling the definition of the gamma factor:

1 γ(v) = √ 1−

|v|2 c2

so that the proper velocity is more compactly:

u = γ(v)(1 + v) The proper velocity is a positive unimodular paravector, which implies the following condition in terms of the Clifford conjugation

u¯ u=1 The proper velocity transforms under the action of the Lorentz rotor L as

u → u′ = LuL† .

7.1.4

Four-momentum paravector

The four-momentum in APS can be obtained by multiplying the proper velocity with the mass as

7.2. CLASSICAL ELECTRODYNAMICS

17

p = mu, with the mass shell condition translated into

p¯p = m2

7.2 Classical electrodynamics Main article: Classical electrodynamics

7.2.1

The electromagnetic field, potential and current

The electromagnetic field is represented as a bi-paravector F:

F = E + iB with the Hermitian part representing the electric field E and the anti-Hermitian part representing the magnetic field B. In the standard Pauli matrix representation, the electromagnetic field is: ( F →

E3 E1 + iE2

E1 − iE2 −E3

)

( B3 +i B1 + iB2

) B1 − iB2 . −B3

The source of the field F is the electromagnetic four-current:

j = ρ + j, where the scalar part equals the electric charge density ρ, and the vector part the electric current density j. Introducing the electromagnetic potential paravector defined as:

A = ϕ + A, in which the scalar part equals the electric potential ϕ, and the vector part the magnetic potential A. The electromagnetic field is then also:

¯V. F = ⟨∂ A⟩ and F is invariant under a gauge transformation of the form

A → A + ∂χ , where χ is a scalar field. The electromagnetic field is covariant under Lorentz transformations according to the law

¯. F → F ′ = LF L

18

CHAPTER 7. ALGEBRA OF PHYSICAL SPACE

7.2.2

Maxwell’s equations and the Lorentz force

The Maxwell equations can be expressed in a single equation: ¯ = 1 ¯j , ∂F ϵ where the overbar represents the Clifford conjugation. The Lorentz force equation takes the form dp = e⟨F u⟩R . dτ

7.2.3

Electromagnetic Lagrangian

The electromagnetic Lagrangian is

L=

1 ⟨F F ⟩S − ⟨A¯j⟩S , 2

which is a real scalar invariant.

7.3 Relativistic quantum mechanics Main article: Relativistic quantum mechanics The Dirac equation, for an electrically charged particle of mass m and charge e, takes the form: ¯ 3 + eAΨ ¯ = mΨ ¯† i∂Ψe where e3 is an arbitrary unitary vector, and A is the electromagnetic paravector potential as above. The electromagnetic interaction has been included via minimal coupling in terms of the potential A.

7.4 Classical spinor Main article: spinor The differential equation of the Lorentz rotor that is consistent with the Lorentz force is dΛ e = F Λ, dτ 2mc such that the proper velocity is calculated as the Lorentz transformation of the proper velocity at rest u = ΛΛ† , which can be integrated to find the space-time trajectory x(τ ) with the additional use of dx =u dτ

7.5. SEE ALSO

19

7.5 See also • Paravector • Multivector • wikibooks:Physics in the Language of Geometric Algebra. An Approach with the Algebra of Physical Space • Dirac equation in the algebra of physical space

7.6 References 7.6.1

Textbooks

• Baylis, William (2002). Electrodynamics: A Modern Geometric Approach (2nd ed.). Birkhäuser. ISBN 08176-4025-8 • W. E. Baylis, editor, Clifford (Geometric) Algebra with Applications to Physics, Mathematics, and Engineering, Birkhäuser, Boston 1996. • Chris Doran and Anthony Lasenby, Geometric Algebra for Physicists, Cambridge University Press (2003) • David Hestenes: New Foundations for Classical Mechanics (Second Edition). ISBN 0-7923-5514-8, Kluwer Academic Publishers (1999)

7.6.2

Articles

• Baylis, William (2002). Relativity in Introductory Physics, Can. J. Phys. 82 (11), 853—873 (2004). (ArXiv: physics/0406158) • W. E. Baylis and G. Jones, The Pauli-Algebra Approach to Special Relativity, J. Phys. A22, 1-16 (1989) • W. E. Baylis, Classical eigenspinors and the Dirac equation, Phys Rev. A, Vol 45, number 7 (1992) • W. E. Baylis, Relativistic dynamics of charges in electromagnetic fields: An eigenspinor approach, Phys Rev. A, Vol 60, number 2 (1999)

Chapter 8

Algebra representation In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint functors page); there is no essential difference between modules for the resulting unital ring, in which the identity acts by the identity mapping, and representations of the algebra.

8.1 Examples 8.1.1

Linear complex structure

Main article: Linear complex structure One of the simplest non-trivial examples is a linear complex structure, which is a representation of the complex numbers C, thought of as an associative algebra over the real numbers R. This algebra is realized concretely as C = R[i]/(i2 + 1), which corresponds to i2 = −1 . Then a representation of C is a real vector space V, together with an action of C on V (a map C → End(V ) ). Concretely, this is just an action of i , as this generates the algebra, and the operator representing i (the image of i in End(V)) is denoted J to avoid confusion with the identity matrix I).

8.1.2

Polynomial algebras

Another important basic class of examples are representations of polynomial algebras, the free commutative algebras – these form a central object of study in commutative algebra and its geometric counterpart, algebraic geometry. A representation of a polynomial algebra in k variables over the field K is concretely a K-vector space with k commuting operators, and is often denoted K[T1 , . . . , Tk ], meaning the representation of the abstract algebra K[x1 , . . . , xk ] where xi 7→ Ti . A basic result about such representations is that, over an algebraically closed field, the representing matrices are simultaneously triangularisable. Even the case of representations of the polynomial algebra in a single variable are of interest – this is denoted by K[T ] and is used in understanding the structure of a single linear operator on a finite-dimensional vector space. Specifically, applying the structure theorem for finitely generated modules over a principal ideal domain to this algebra yields as corollaries the various canonical forms of matrices, such as Jordan canonical form. In some approaches to noncommutative geometry, the free noncommutative algebra (polynomials in non-commuting variables) plays a similar role, but the analysis is much more difficult.

8.2 Weights Main article: Weight (representation theory)

20

8.3. SEE ALSO

21

Eigenvalues and eigenvectors can be generalized to algebra representations. The generalization of an eigenvalue of an algebra representation is, rather than a single scalar, a one-dimensional representation λ : A → R (i.e., an algebra homomorphism from the algebra to its underlying ring: a linear functional that is also multiplicative).[note 1] This is known as a weight, and the analog of an eigenvector and eigenspace are called weight vector and weight space. The case of the eigenvalue of a single operator corresponds to the algebra R[T ], and a map of algebras R[T ] → R is determined by which scalar it maps the generator T to. A weight vector for an algebra representation is a vector such that any element of the algebra maps this vector to a multiple of itself – a one-dimensional submodule (subrepresentation). As the pairing A × M → M is bilinear, “which multiple” is an A-linear functional of A (an algebra map A → R), namely the weight. In symbols, a weight vector is a vector m ∈ M such that am = λ(a)m for all elements a ∈ A, for some linear functional λ – note that on the left, multiplication is the algebra action, while on the right, multiplication is scalar multiplication. Because a weight is a map to a commutative ring, the map factors through the abelianization of the algebra A – equivalently, it vanishes on the derived algebra – in terms of matrices, if v is a common eigenvector of operators T and U , then T U v = U T v (because in both cases it is just multiplication by scalars), so common eigenvectors of an algebra must be in the set on which the algebra acts commutatively (which is annihilated by the derived algebra). Thus of central interest are the free commutative algebras, namely the polynomial algebras. In this particularly simple and important case of the polynomial algebra F[T1 , . . . , Tk ] in a set of commuting matrices, a weight vector of this algebra is a simultaneous eigenvector of the matrices, while a weight of this algebra is simply a k -tuple of scalars λ = (λ1 , . . . , λk ) corresponding to the eigenvalue of each matrix, and hence geometrically to a point in k -space. These weights – in particularly their geometry – are of central importance in understanding the representation theory of Lie algebras, specifically the finite-dimensional representations of semisimple Lie algebras. As an application of this geometry, given an algebra that is a quotient of a polynomial algebra on k generators, it corresponds geometrically to an algebraic variety in k -dimensional space, and the weight must fall on the variety – i.e., it satisfies defining equations for the variety. This generalizes the fact that eigenvalues satisfy the characteristic polynomial of a matrix in one variable.

8.3 See also • Representation theory • Intertwiner • Representation theory of Hopf algebras • Lie algebra representation • Schur’s lemma • Jacobson density theorem • Double commutant theorem

8.4 Notes [1] Note that for a field, the endomorphism algebra of a one-dimensional vector space (a line) is canonically equal to the underlying field: End(L) = K, since all endomorphisms are scalar multiplication; there is thus no loss in restricting to concrete maps to the base field, rather than to abstract 1-dimensional representations. For rings there are also maps to quotients rings, which need not factor through maps to the ring itself, but again abstract 1-dimensional modules are not needed.

8.5 References

Chapter 9

Algebraically compact module In mathematics, especially in the area of abstract algebra known as module theory, algebraically compact modules, also called pure-injective modules, are modules that have a certain “nice” property which allows the solution of infinite systems of equations in the module by finitary means. The solutions to these systems allow the extension of certain kinds of module homomorphisms. These algebraically compact modules are analogous to injective modules, where one can extend all module homomorphisms. All injective modules are algebraically compact, and the analogy between the two is made quite precise by a category embedding.

9.1 Definitions Suppose R is a ring and M is a left R-module. Take two sets I and J, and for every i in I and j in J, an element rij of R such that, for every i in I, only finitely many rij are non-zero. Furthermore, take an element mi of M for every i in I. These data describe a system of linear equations in M: ∑ j∈J

rij xj = mi for every i∈I.

The goal is to decide whether this system has a solution, i.e. whether there exist elements xj of M for every j in J such that all the equations of the system are simultaneously satisfied. (Note that we do not require that only finitely many of the xj are non-zero here.) Now consider such a system of linear equations, and assume that any subsystem consisting of only finitely many equations is solvable. (The solutions to the various subsystems may be different.) If every such “finitely-solvable” system is itself solvable, then we call the module M algebraically compact. A module homomorphism M → K is called pure injective if the induced homomorphism between the tensor products C ⊗ M → C ⊗ K is injective for every right R-module C. The module M is pure-injective if any pure injective homomorphism j : M → K splits (i.e. there exists f : K → M with fj = 1M). It turns out that a module is algebraically compact if and only if it is pure-injective.

9.2 Examples Every vector space is algebraically compact (since it is pure-injective). More generally, every injective module is algebraically compact, for the same reason. If R is an associative algebra with 1 over some field k, then every R-module with finite k-dimension is algebraically compact. This gives rise to the intuition that algebraically compact modules are those (possibly “large”) modules which share the nice properties of “small” modules. The Prüfer groups are algebraically compact abelian groups (i.e. Z-modules). Many algebraically compact modules can be produced using the injective cogenerator Q/Z of abelian groups. If H is a right module over the ring R, one forms the (algebraic) character module H* consisting of all group homomorphisms 22

9.3. FACTS

23

from H to Q/Z. This is then a left R-module, and the *-operation yields a faithful contravariant functor from right R-modules to left R-modules. Every module of the form H* is algebraically compact. Furthermore, there are pure injective homomorphisms H → H**, natural in H. One can often simplify a problem by first applying the *-functor, since algebraically compact modules are easier to deal with.

9.3 Facts The following condition is equivalent to M being algebraically compact: • For every index set I, the addition map M(I) → M can be extended to a module homomorphism MI → M (here M(I) denotes the direct sum of copies of M, one for each element of I; MI denotes the product of copies of M, one for each element of I). Every indecomposable algebraically compact module has a local endomorphism ring. Algebraically compact modules share many other properties with injective objects because of the following: there exists an embedding of R-Mod into a Grothendieck category G under which the algebraically compact R-modules precisely correspond to the injective objects in G.

9.4 See also • Table of mathematical symbols

9.5 References • C.U. Jensen and H. Lenzing: Model Theoretic Algebra, Gordon and Breach, 1989

Chapter 10

Almost commutative ring In algebra, a filtered ring A is said to be almost commutative if the associated graded ring gr A = ⊕Ai /Ai−1 is commutative. Basic examples of almost commutative rings involve differential operators. For example, the enveloping algebra of a complex Lie algebra is almost commutative by the PBW theorem. Similarly, a Weyl algebra is almost commutative.

10.1 See also • Ore condition • Gelfand–Kirillov dimension

10.2 References • Victor Ginzburg, Lectures on D-modules

24

Chapter 11

Annihilator (ring theory) In mathematics, specifically module theory, the annihilator of a set is a concept generalizing torsion and orthogonality.

11.1 Definitions Let R be a ring, and let M be a left R-module. Choose a nonempty subset S of M. The annihilator, denoted AnnR(S), of S is the set of all elements r in R such that for each s in S, rs = 0.[1] In set notation, AnnR (S) = {r ∈ R | ∀s ∈ S, rs = 0} It is the set of all elements of R that “annihilate” S (the elements for which S is torsion). Subsets of right modules may be used as well, after the modification of "sr = 0” in the definition. The annihilator of a single element x is usually written AnnR(x) instead of AnnR({x}). If the ring R can be understood from the context, the subscript R can be omitted. Since R is a module over itself, S may be taken to be a subset of R itself, and since R is both a right and a left R module, the notation must be modified slightly to indicate the left or right side. Usually ℓ.AnnR (S) and r.AnnR (S) or some similar subscript scheme are used to distinguish the left and right annihilators, if necessary. If M is an R-module and AnnR(M) = 0, then M is called a faithful module.

11.2 Properties If S is a subset of a left R module M, then Ann(S) is a left ideal of R. The proof is straightforward: If a and b both annihilate S, then for each s in S, (a + b)s = as + bs = 0, and for any r in R, (ra)s = r(as) = r0 = 0. (A similar proof follows for subsets of right modules to show that the annihilator is a right ideal.) If S is a submodule of M, then AnnR(S) is even a two-sided ideal: (ac)s = a(cs) = 0, since cs is another element of S.[2] If S is a subset of M and N is the submodule of M generated by S, then in general AnnR(N) is a subset of AnnR(S), but they are not necessarily equal. If R is commutative, then it is easy to check that equality holds. M may be also viewed as a R/AnnR(M)-module using the action rm := rm . Incidentally, it is not always possible to make an R module into an R/I module this way, but if the ideal I is a subset of the annihilator of M, then this action is well defined. Considered as an R/AnnR(M)-module, M is automatically a faithful module.

11.3 Chain conditions on annihilator ideals The lattice of ideals of the form ℓ.AnnR (S) where S is a subset of R comprise a complete lattice when partially ordered by inclusion. It is interesting to study rings for which this lattice (or its right counterpart) satisfy the ascending chain 25

26

CHAPTER 11. ANNIHILATOR (RING THEORY)

condition or descending chain condition. Denote the lattice of left annihilator ideals of R as LA and the lattice of right annihilator ideals of R as RA . It is known that LA satisfies the A.C.C. if and only if RA satisfies the D.C.C., and symmetrically RA satisfies the A.C.C. if and only if LA satisfies the D.C.C. If either lattice has either of these chain conditions, then R has no infinite orthogonal sets of idempotents. (Anderson 1992, p.322) (Lam 1999) If R is a ring for which LA satisfies the A.C.C. and RR has finite uniform dimension, then R is called a left Goldie ring. (Lam 1999)

11.4 Category-theoretic description for commutative rings When R is commutative and M is an R-module, we may describe AnnR(M) as the kernel of the action map R → EndR(M) determined by the adjunct map of the identity M → M along the Hom-tensor adjunction. More generally, given a bilinear map of modules F : M × N → P , the annihilator of a subset S ⊂ M is the set of all elements in N that annihilate S :

Ann(S) := {n ∈ N | ∀s ∈ S, F (s, n) = 0}. Conversely, given T ⊂ N , one can define an annihilator as a subset of M . The annihilator gives a Galois connection between subsets of M and N , and the associated closure operator is stronger than the span. In particular: • annihilators are submodules • Span(S) ≤ Ann(Ann(S)) • Ann(Ann(Ann(S))) = Ann(S) An important special case is in the presence of a nondegenerate form on a vector space, particularly an inner product: then the annihilator associated to the map V × V → K is called the orthogonal complement.

11.5 Relations to other properties of rings • Annihilators are used to define left Rickart rings and Baer rings. • The set of (left) zero divisors DS of S can be written as

DS =

∪

AnnR (x).

x∈S, x̸=0

(Here we allow zero to be a zero divisor.) In particular DR is the set of (left) zero divisors of R taking S = R and R acting on itself as a left R-module. • When R is commutative and Noetherian, the set DR is precisely equal to the union of the associated prime ideals of R.

11.6 See also • socle

11.7. NOTES

27

11.7 Notes [1] Pierce (1982), p. 23. [2] Pierce (1982), p. 23, Lemma b, item (i).

11.8 References • Weisstein, Eric W., “Annihilator”, MathWorld. • Anderson, Frank W.; Fuller, Kent R. (1992), Rings and categories of modules, Graduate Texts in Mathematics 13 (2 ed.), New York: Springer-Verlag, pp. x+376, ISBN 0-387-97845-3, MR 1245487 • Israel Nathan Herstein (1968) Noncommutative Rings, Carus Mathematical Monographs #15, Mathematical Association of America, page 3. • Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, pp. 228–232, ISBN 978-0-387-98428-5, MR 1653294 • Richard S. Pierce. Associative algebras. Graduate texts in mathematics, Vol. 88, Springer-Verlag, 1982, ISBN 978-0-387-90693-5

Chapter 12

Arithmetical ring In algebra, a commutative ring R is said to be arithmetical (or arithmetic) if any of the following equivalent conditions holds: 1. The localization Rm of R at m is a uniserial ring for every maximal ideal m of R. 2. For all ideals a, b , and c , a ∩ (b + c) = (a ∩ b) + (a ∩ c) 3. For all ideals a, b , and c , a + (b ∩ c) = (a + b) ∩ (a + c) The last two conditions both say that the lattice of all ideals of R is distributive. An arithmetical domain is the same thing as a Prüfer domain.

12.1 References

• Boynton, Jason (2007). “Pullbacks of arithmetical rings”. Commun. Algebra 35 (9): 2671–2684. doi:10.1080/009278707013512 ISSN 0092-7872. Zbl 1152.13015. • Fuchs, Ladislas (1949). "Über die Ideale arithmetischer Ringe”. Comment. Math. Helv. (in German) 23: 334–341. doi:10.1007/bf02565607. ISSN 0010-2571. Zbl 0040.30103. • Larsen, Max D.; McCarthy, Paul Joseph (1971). Multiplicative theory of ideals. Pure and Applied Mathematics 43. Academic Press. pp. 150–151. ISBN 0080873561. Zbl 0237.13002.

12.2 External links Arithmetical ring at PlanetMath.org.

28

Chapter 13

Artin algebra In algebra, an Artin algebra is an algebra Λ over a commutative Artin ring R that is a finitely generated R-module. They are named after Emil Artin. Every Artin algebra is an Artin ring.

13.1 Dual and transpose There are several different dualities taking finitely generated modules over Λ to modules over the opposite algebra Λop . • If M is a left Λ module then the right Λ-module M * is defined to be HomΛ(M,Λ). • The dual D(M) of a left Λ-module M is the right Λ-module D(M) = HomR(M,J), where J is the dualizing module of R, equal to the sum of the injective envelopes of the non-isomorphic simple R-modules or equivalently the injective envelope of R/rad R. The dual of a left module over Λ does not depend on the choice of R (up to isomorphism). • The transpose Tr(M) of a left Λ-module M is a right Λ-module defined to be the cokernel of the map Q* → P * , where P → Q → M → 0 is a minimal projective presentation of M.

13.2 References • Auslander, Maurice; Reiten, Idun; Smalø, Sverre O. (1997) [1995], Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, ISBN 978-0-521-59923-8, MR 1314422, Zbl 0834.16001

29

Chapter 14

Artinian ideal In abstract algebra, an Artinian ideal, named after Emil Artin, is encountered in ring theory, in particular, with polynomial rings. Given a polynomial ring R = k[X1 , ... Xn] where k is some field, an Artinian ideal is an ideal I in R for which the Krull dimension of the quotient ring R/I is 0. Also, less precisely, one can think of an Artinian ideal as one that has at least each indeterminate in R raised to a power greater than 0 as a generator. If an ideal is not Artinian, one can take the Artinian closure of it as follows. First, take the least common multiple of the generators of the ideal. Second, add to the generating set of the ideal each indeterminate of the LCM with its power increased by 1 if the power is not 0 to begin with. An example is below.

14.1 Examples Let R = k[x, y, z] , and let I = (x2 , y 5 , z 4 ), J = (x3 , y 2 , z 6 , x2 yz 4 , yz 3 ) and K = (x3 , y 4 , x2 z 7 ) . Here, I and J are Artinian ideals, but K is not because in K , the indeterminate z does not appear alone to a power as a generator. ˆ , we find the LCM of the generators of K , which is x3 y 4 z 7 . Then, we add To take the Artinian closure of K , K 4 5 8 ˆ = (x3 , y 4 , z 8 , x2 z 7 ) which is Artinian. the generators x , y , and z to K , and reduce. Thus, we have K

14.2 References • Sáenz-de-Cabezón Irigaray, Eduardo. “Combinatorial Koszul Homology, Computations and Applications”. arXiv:0803.0421.

30

Chapter 15

Artinian module In abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if it is an Artinian module over itself (with left or right multiplication). Both concepts are named for Emil Artin. In the presence of the axiom of choice, the descending chain condition becomes equivalent to the minimum condition, and so that may be used in the definition instead. Like Noetherian modules, Artinian modules enjoy the following heredity property: • If M is an Artinian R-module, then so is any submodule and any quotient of M. The converse also holds: • If M is any R module and N any Artinian submodule such that M/N is Artinian, then M is Artinian. As a consequence, any finitely-generated module over an Artinian ring is Artinian.[1] Since an Artinian ring is also a Noetherian ring, and finitely-generated modules over a Noetherian ring are Noetherian,[1] it is true that for an Artinian ring R, any finitely-generated R-module is both Noetherian and Artinian, and is said to be of finite length; however, if R is not Artinian, or if M is not finitely generated, there are counterexamples.

15.1 Left and right Artinian rings, modules and bimodules The ring R can be considered as a right module, where the action is the natural one given by the ring multiplication on the right. R is called right Artinian when this right module R is an Artinian module. The definition of “left Artinian ring” is done analogously. For noncommutative rings this distinction is necessary, because it is possible for a ring to be Artinian on one side only. The left-right adjectives are not normally necessary for modules, because the module M is usually given as a left or right R module at the outset. However, it is possible that M may have both a left and right R module structure, and then calling M Artinian is ambiguous, and it becomes necessary to clarify which module structure is Artinian. To separate the properties of the two structures, one can abuse terminology and refer to M as left Artinian or right Artinian when, strictly speaking, it is correct to say that M, with its left R-module structure, is Artinian. The occurrence of modules with a left and right structure is not unusual: for example R itself has a left and right R module structure. In fact this is an example of a bimodule, and it may be possible for an abelian group M to be made into a left-R, right-S bimodule for a different ring S. Indeed, for any right module M, it is automatically a left module over the ring of integers Z, and moreover is a Z-R bimodule. For example, consider the rational numbers Q as a Z-Q bimodule in the natural way. Then Q is not Artinian as a left Z module, but it is Artinian a right Q module. The Artinian condition can be defined on bimodule structures as well: an Artinian bimodule is a bimodule whose poset of sub-bimodules satisfies the descending chain condition. Since a sub-bimodule of an R-S bimodule M is a fortiori a left R-module, if M considered as a left R module were Artinian, then M is automatically an Artinian bimodule. It may happen, however, that a bimodule is Artinian without its left or right structures being Artinian, as the following example will show. 31

32

CHAPTER 15. ARTINIAN MODULE

Example: It is well known that a simple ring is left Artinian if and only if it is right Artinian, in which case it is a semisimple ring. Let R be a simple ring which is not right Artinian. Then it is also not left Artinian. Considering R as an R-R bimodule in the natural way, its sub-bimodules are exactly the ideals of R. Since R is simple there are only two: R and the zero ideal. Thus the bimodule R is Artinian as a bimodule, but not Artinian as a left or right R-module over itself.

15.2 Relation to the Noetherian condition Unlike the case of rings, there are Artinian modules which are not Noetherian modules. For example, consider the p-primary component of Q/Z , that is Z[1/p]/Z , which is isomorphic to the p-quasicyclic group Z(p∞ ) , regarded as Z -module. The chain ⟨1/p⟩ ⊂ ⟨1/p2 ⟩ ⊂ ⟨1/p3 ⟩ ⊂ · · · does not terminate, so Z(p∞ ) (and therefore Q/Z ) is not Noetherian. Yet every descending chain of (without loss of generality) proper submodules terminates: Each such chain has the form ⟨1/n1 ⟩ ⊇ ⟨1/n2 ⟩ ⊇ ⟨1/n3 ⟩ ⊇ · · · for some integers n1 , n2 , n3 , . . . , and the inclusion of ⟨1/ni+1 ⟩ ⊆ ⟨1/ni ⟩ implies that ni+1 must divide ni . So n1 , n2 , n3 , . . . is a decreasing sequence of positive integers. Thus the sequence terminates, making Z(p∞ ) Artinian. Over a commutative ring, every cyclic Artinian module is also Noetherian, but over noncommutative rings cyclic Artinian modules can have uncountable length as shown in the article of Hartley and summarized nicely in the Paul Cohn article dedicated to Hartley’s memory.

15.3 See also • Noetherian module • Ascending/Descending chain condition • Composition series • Krull dimension

15.4 Notes [1] Lam (2001), Proposition 1.21, p. 19.

15.5 References • Atiyah, M.F.; Macdonald, I.G. (1969). “Chapter 6. Chain conditions; Chapter 8. Artin rings”. Introduction to Commutative Algebra. Westview Press. ISBN 978-0-201-40751-8. • Cohn, P.M. (1997). “Cyclic Artinian Modules Without a Composition Series”. J. London Math. Soc. (2) 55 (2): 231–235. doi:10.1112/S0024610797004912. MR 1438626. • Hartley, B. (1977). “Uncountable Artinian modules and uncountable soluble groups satisfying Min-n”. Proc. London Math. Soc. (3) 35 (1): 55–75. doi:10.1112/plms/s3-35.1.55. MR 442091. • Lam, T.Y. (2001). “Chapter 1. Wedderburn-Artin theory”. A First Course in Noncommutative Rings. Springer Verlag. ISBN 978-0-387-95325-0.

Chapter 16

Artinian ring In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition. A ring is left Artinian if it satisfies the descending chain condition on left ideals, right Artinian if it satisfies the descending chain condition on right ideals, and Artinian or two-sided Artinian if it is both left and right Artinian. For commutative rings the left and right definitions coincide, but in general they are distinct from each other. The Artin–Wedderburn theorem characterizes all simple Artinian rings as the matrix rings over a division ring. This implies that a simple ring is left Artinian if and only if it is right Artinian. Although the descending chain condition appears dual to the ascending chain condition, in rings it is in fact the stronger condition. Specifically, a consequence of the Akizuki–Hopkins–Levitzki theorem is that a left (resp. right) Artinian ring is automatically a left (resp. right) Noetherian ring. This is not true for general modules, that is, an Artinian module need not be a Noetherian module.

16.1 Examples • An integral domain is Artinian if and only if it is a field. • A ring with finitely many, say left, ideals is left Artinian. In particular, a finite ring (e.g., Z/nZ ) is left and right Artinian. • Let k be a field. Then k[t]/(tn ) is Artinian for every positive integer n. • If I is a nonzero ideal of a Dedekind domain A, then A/I is a principal Artinian ring.[1] • For each n ≥ 1 , the full matrix ring Mn (R) over a left Artinian (resp. left Noetherian) ring R is left Artinian (resp. left Noetherian).[2] The ring of integers Z is a Noetherian ring but is not Artinian.

16.2 Modules over Artinian rings Let M be a left module over a left Artinian ring. Then the following are equivalent (Hopkins’ theorem): (i) M is finitely generated, (ii) M has finite length (i.e., has composition series), (iii) M is Noetherian, (iv) M is Artinian.[3]

16.3 Commutative Artinian rings Let A be a commutative Noetherian ring with unity. Then the following are equivalent. 33

34

CHAPTER 16. ARTINIAN RING • A is Artinian. • A is a finite product of commutative Artinian local rings.[4] • A / nil(A) is a semisimple ring, where nil(A) is the nilradical of A.[5] • Every finitely generated module over A has finite length. (see above) • A has Krull dimension zero.[6] (In particular, the nilradical is the Jacobson radical since prime ideals are maximal.) • Spec A is finite and discrete. • Spec A is discrete.[7]

Let k be a field and A finitely generated k-algebra. Then A is Artinian if and only if A is finitely generated as k-module. An Artinian local ring is complete. A quotient and localization of an Artinian ring is Artinian.

16.4 Simple Artinian ring A simple Artinian ring A is a matrix ring over a division ring. Indeed,[8] let I be a minimal (nonzero) right ideal of A. Then, since AI is a two-sided ideal, AI = A since A is simple. Thus, we can choose ai ∈ A so that 1 ∈ a1 I + · · · + ak I . Assume k is minimal with respect that property. Consider the map of right A-modules: I ⊕k → A, (y1 , . . . , yk ) 7→ a1 y1 + · · · + ak yk . It is surjective. If it is not injective, then, say, a1 y1 = a2 y2 + · · · + ak yk with nonzero y1 . Then, by the minimality of I, we have: y1 A = I . It follows:

a1 I = a1 y1 A ⊂ a2 I + · · · + ak I which contradicts the minimality of k. Hence, I ⊕k ≃ A and thus A ≃ EndA (A) ≃ Mk (EndA (I)) .

16.5 See also • Artin algebra • Artinian ideal • Serial module • Semiperfect ring • Noetherian ring

16.6 Notes [1] Theorem 459 of http://math.uga.edu/~{}pete/integral.pdf [2] Cohn 2003, 5.2 Exercise 11 [3] Bourbaki, VIII, pg 7 [4] Atiyah & Macdonald 1969, Theorems 8.7 [5] Sketch: In commutative rings, nil(A) is contained in the Jacobson radical of A. Since A/nil(A) is semisimple, nil(A) is actually equal to the Jacobson radical of A. By Levitzky’s theorem, nil(A) is a nilpotent ideal. These last two facts show that A is a semiprimary ring, and by the Hopkins–Levitzki theorem A is Artinian.

16.7. REFERENCES

35

[6] Atiyah & Macdonald 1969, Theorems 8.5 [7] Atiyah & Macdonald 1969, Ch. 8, Exercise 2. [8] Milnor, John Willard (1971), Introduction to algebraic K-theory, Annals of Mathematics Studies 72, Princeton, NJ: Princeton University Press, p. 144, MR 0349811, Zbl 0237.18005

16.7 References • Auslander, Maurice; Reiten, Idun; Smalø, Sverre O. (1995), Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, doi:10.1017/CBO9780511623608, ISBN 978-0-521-41134-9, MR 1314422 • Bourbaki, Algèbre • Charles Hopkins. Rings with minimal condition for left ideals. Ann. of Math. (2) 40, (1939). 712–730. • Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8 • Cohn, Paul Moritz (2003). Basic algebra: groups, rings, and fields. Springer. ISBN 978-1-85233-587-8.

Chapter 17

Artin–Rees lemma In mathematics, the Artin–Rees lemma is a basic result about modules over a Noetherian ring, along with results such as the Hilbert basis theorem. It was proved in the 1950s in independent works by the mathematicians Emil Artin and David Rees; a special case was known to Oscar Zariski prior to their work. One consequence of the lemma is the Krull intersection theorem. The result is also used to prove the exactness property of completion (Atiyah & MacDonald 1969, pp. 107–109).

17.1 Statement Let I be an ideal in a Noetherian ring R; let M be a finitely generated R-module and let N a submodule of M. Then there exists an integer k ≥ 1 so that, for n ≥ k, I n M ∩ N = I n−k ((I k M ) ∩ N ).

17.2 Proof The lemma immediately follows from the fact that R is “Noetherian” once necessary notions and notations are set up.[1] ⊕∞ n For any ring R and an ideal I in R, we set BI R = n=0 I (B for blow-up.) We say a decreasing sequence of submodules M = M0 ⊃ M1 ⊃ M2 ⊃ · · · is an I-filtration if IMn ⊂ n+1 ; moreover, it is stable if IMn = Mn+1 ⊕M ∞ for sufficiently large n. If M is given an I-filtration, we set BI M = n=0 Mn ; it is a graded module over BI R . Now, let M be a R-module with the I-filtration Mi by finitely generated R-modules. We make an observation BI M is a finitely generated module over BI R if and only if the filtration is I-stable. Indeed, if the filtration is I-stable, then BI M is generated by the first k + 1 terms M0 , . . . , Mk and those terms are ⊕k finitely generated; thus, BI M is finitely generated. Conversely, if it is finitely generated, say, by j=0 Mj , then, for n ≥ k , each f in Mn can be written as

f=

∑

aij gij ,

aij ∈ I n−j

with the generators gij in Mj , j ≤ k . That is, f ∈ I n−k Mk . We can now prove the lemma, assuming R is Noetherian. Let Mn = I n M . Then Mn are an I-stable filtration. Thus, by the observation, BI M is finitely generated over BI R . But BI R ≃ R[It] is a Noetherian ring since R is. (The ring R[It] is called the Rees algebra.) Thus, BI M is a Noetherian module and any submodule is finitely generated over BI R ; in particular, BI N is finitely generated when N is given the induced filtration; i.e., Nn = Mn ∩ N . Then the induced filtration is I-stable again by the observation. 36

17.3. PROOF OF KRULL’S INTERSECTION THEOREM

37

17.3 Proof of Krull’s intersection theorem Besides the use in completion of a ring, a typical application of the lemma is the proof of the Krull’s intersection ∩∞ theorem, which says: n=1 I n = 0 for a proper ideal I in a Noetherian local ring. By the lemma applied to the intersection N, we find k such that for n ≥ k ,

I n ∩ N = I n−k (I k ∩ N ). But then N = IN and thus N = 0 by Nakayama.

17.4 References [1] Eisenbud, Lemma 5.1

• Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8 • Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8.

17.5 External links • Artin-Rees Theorem at PlanetMath.org.

Chapter 18

Artin–Wedderburn theorem In abstract algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings and semisimple algebras. The theorem states that an (Artinian) [1] semisimple ring R is isomorphic to a product of finitely many ni-byni matrix rings over division rings Di, for some integers ni, both of which are uniquely determined up to permutation of the index i. In particular, any simple left or right Artinian ring is isomorphic to an n-by-n matrix ring over a division ring D, where both n and D are uniquely determined.[2] As a direct corollary, the Artin–Wedderburn theorem implies that every simple ring that is finite-dimensional over a division ring (a simple algebra) is a matrix ring. This is Joseph Wedderburn's original result. Emil Artin later generalized it to the case of Artinian rings. Note that if R is a finite-dimensional simple algebra over a division ring E, D need not be contained in E. For example, matrix rings over the complex numbers are finite-dimensional simple algebras over the real numbers. The Artin–Wedderburn theorem reduces classifying simple rings over a division ring to classifying division rings that contain a given division ring. This in turn can be simplified: The center of D must be a field K. Therefore R is a K-algebra, and itself has K as its center. A finite-dimensional simple algebra R is thus a central simple algebra over K. Thus the Artin–Wedderburn theorem reduces the problem of classifying finite-dimensional central simple algebras to the problem of classifying division rings with given center.

18.1 Examples Let R be the field of real numbers, C be the field of complex numbers, and H the quaternions. • Every finite-dimensional simple algebra over R must be a matrix ring over R, C, or H. Every central simple algebra over R must be a matrix ring over R or H. These results follow from the Frobenius theorem. • Every finite-dimensional simple algebra over C must be a matrix ring over C and hence every central simple algebra over C must be a matrix ring over C. • Every finite-dimensional central simple algebra over a finite field must be a matrix ring over that field. • Every commutative semisimple ring must be a finite direct product of fields.[3] • The Artin–Wedderburn theorem implies that a semisimple algebra over a field k is isomorphic to a finite ∏ product Mni (Di ) where the ni are natural numbers, the Di are finite dimensional division algebras over k , and Mni (Di ) is the algebra of ni × ni matrices over Di . Again, this product is unique up to permutation of the factors.

18.2 See also • Maschke’s theorem • Brauer group 38

18.3. REFERENCES

39

• Jacobson density theorem • Hypercomplex number

18.3 References [1] Semisimple rings are necessarily Artinian rings. Some authors use “semisimple” to mean the ring has a trivial Jacobson radical. For Artinian rings, the two notions are equivalent, so “Artinian” is included here to eliminate that ambiguity. [2] John A. Beachy (1999). Introductory Lectures on Rings and Modules. Cambridge University Press. p. 156. ISBN 978-0521-64407-5. [3] This is clear since matrix rings larger than 1×1 are never commutative.

• P. M. Cohn (2003) Basic Algebra: Groups, Rings, and Fields, pages 137–9. • J.H.M. Wedderburn (1908). “On Hypercomplex Numbers”. Proceedings of the London Mathematical Society 6: 77–118. doi:10.1112/plms/s2-6.1.77. • Artin, E. (1927). “Zur Theorie der hyperkomplexen Zahlen” 5. pp. 251–260.

Chapter 19

Artin–Zorn theorem In mathematics, the Artin–Zorn theorem, named after Emil Artin and Max Zorn, states that any finite alternative division ring is necessarily a finite field. It was first published by Zorn, but in his publication Zorn credited it to Artin.[1][2] The Artin–Zorn theorem is a generalization of the Wedderburn theorem, which states that finite associative division rings are fields. As a geometric consequence, every finite Moufang plane is the classical projective plane over a finite field.[3][4]

19.1 References [1] Zorn, M. (1930), “Theorie der alternativen Ringe”, Abh. Math. Sem. Hamburg 8: 123–147. [2] Lüneburg, Heinz (2001), “On the early history of Galois fields”, in Jungnickel, Dieter; Niederreiter, Harald, Finite fields and applications: proceedings of the Fifth International Conference on Finite Fields and Applications Fq5, held at the University of Augsburg, Germany, August 2–6, 1999, Springer-Verlag, pp. 341–355, ISBN 978-3-540-41109-3, MR 1849100. [3] Shult, Ernest (2011), Points and Lines: Characterizing the Classical Geometries, Universitext, Springer-Verlag, p. 123, ISBN 978-3-642-15626-7. [4] McCrimmon, Kevin (2004), A taste of Jordan algebras, Universitext, Springer-Verlag, p. 34, ISBN 978-0-387-95447-9.

40

Chapter 20

Ascending chain condition on principal ideals In abstract algebra, the ascending chain condition can be applied to the posets of principal left, principal right, or principal two-sided ideals of a ring, partially ordered by inclusion. The ascending chain condition on principal ideals (abbreviated to ACCP) is satisfied if there is no infinite strictly ascending chain of principal ideals of the given type (left/right/two-sided) in the ring, or said another way, every ascending chain is eventually constant. The counterpart descending chain condition may also be applied to these posets, however there is currently no need for the terminology “DCCP” since such rings are already called left or right perfect rings. (See Noncommutative ring section below.) Noetherian rings (e.g. principal ideal domains) are typical examples, but some important non-Noetherian rings also satisfy (ACCP), notably unique factorization domains and left or right perfect rings.

20.1 Commutative rings It is well known that a nonzero nonunit in a Noetherian integral domain factors into irreducibles. The proof of this relies on only (ACCP) not (ACC), so in any integral domain with (ACCP), an irreducible factorization exists. (In other words, any integral domains with (ACCP) are atomic. But the converse is false, as shown in (Grams 1974).) Such a factorization may not be unique; the usual way to establish uniqueness of factorizations uses Euclid’s lemma, which requires factors to be prime rather than just irreducible. Indeed one has the following characterization: let A be an integral domain. Then the following are equivalent. 1. A is a UFD. 2. A satisfies (ACCP) and every irreducible of A is prime. 3. A is a GCD domain satisfying (ACCP). The so-called Nagata criterion holds for an integral domain A satisfying (ACCP): Let S be a multiplicatively closed subset of A generated by prime elements. If the localization S −1 A is a UFD, so is A. (Nagata 1975, Lemma 2.1) (Note that the converse of this is trivial.) An integral domain A satisfies (ACCP) if and only if the polynomial ring A[t] does.[1] The analogous fact is false if A is not an integral domain. (Heinzer, Lantz 1994) An integral domain where every finitely generated ideal is principal (that is, a Bézout domain) satisfies (ACCP) if and only if it is a principal ideal domain.[2] The ring Z+XQ[X] of all rational polynomials with integral constant term is an example of an integral domain (actually a GCD domain) that does not satisfy (ACCP), for the chain of principal ideals

(X) ⊂ (X/2) ⊂ (X/4) ⊂ (X/8), ... 41

42

CHAPTER 20. ASCENDING CHAIN CONDITION ON PRINCIPAL IDEALS

is non-terminating.

20.2 Noncommutative rings In the noncommutative case, it becomes necessary to distinguish the right ACCP from left ACCP. The former only requires the poset of ideals of the form xR to satisfy the ascending chain condition, and the latter only examines the poset of ideals of the form Rx. A theorem of Hyman Bass in (Bass 1960) now known as “Bass’ Theorem P” showed that the descending chain condition on principal left ideals of a ring R is equivalent to R being a right perfect ring. D. Jonah showed in (Jonah 1970) that there is a side-switching connection between the ACCP and perfect rings. It was shown that if R is right perfect (satisfies right DCCP), then R satisfies the left ACCP, and symmetrically, if R is left perfect (satisfies left DCCP), then it satisfies the right ACCP. The converses are not true, and the above switches from “left” and “right” are not typos. Whether the ACCP holds on the right or left side of R, it implies that R has no infinite set of nonzero orthogonal idempotents, and that R is a Dedekind finite ring. (Lam 1999, p.230-231)

20.3 References [1] Gilmer, Robert (1986), “Property E in commutative monoid rings”, Group and semigroup rings (Johannesburg, 1985), North-Holland Math. Stud. 126, Amsterdam: North-Holland, pp. 13–18, MR 860048. [2] Proof: In a Bézout domain the ACCP is equivalent to the ACC on finitely generated ideals, but this is known to be equivalent to the ACC on all ideals. Thus the domain is Noetherian and Bézout, hence a principal ideal domain.

• Bass, Hyman (1960), “Finitistic dimension and a homological generalization of semi-primary rings”, Trans. Amer. Math. Soc. 95: 466–488, doi:10.1090/s0002-9947-1960-0157984-8, ISSN 0002-9947, MR 0157984 • Grams, Anne (1974), “Atomic rings and the ascending chain condition for principal ideals”, Proc. Cambridge Philos. Soc. 75: 321–329, doi:10.1017/s0305004100048532, MR 0340249 • Heinzer, William J.; Lantz, David C. (1994), “ACCP in polynomial rings: a counterexample”, Proc. Amer. Math. Soc. 121 (3): 975–977, doi:10.2307/2160301, ISSN 0002-9939, JSTOR 2160301, MR 1653294 • Jonah, David (1970), “Rings with the minimum condition for principal right ideals have the maximum condition for principal left ideals”, Math. Z. 113: 106–112, doi:10.1007/bf01141096, ISSN 0025-5874, MR 0260779 • Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294 • Nagata, Masayoshi (1975), “Some types of simple ring extensions”, Houston J. Math. 1 (1): 131–136, ISSN 0362-1588, MR 0382248

Chapter 21

Associated graded ring In mathematics, the associated graded ring of a ring R with respect to a proper ideal I is the graded ring:

n n+1 grI R = ⊕∞ n=0 I /I

Similarly, if M is a left R-module, then the associated graded module is the graded module over grI R : n n+1 grI M = ⊕∞ M 0 I M /I

21.1 Basic definitions and properties For a ring R and ideal I, multiplication in grI R is defined as follows: First, consider homogeneous elements a ∈ I i /I i+1 and b ∈ I j /I j+1 and suppose a′ ∈ I i is a representative of a and b′ ∈ I j is a representative of b. Then define ab to be the equivalence class of a′ b′ in I i+j /I i+j+1 . Note that this is well-defined modulo I i+j+1 . Multiplication of inhomogeneous elements is defined by using the distributive property. A ring or module may be related to its associated graded through the initial form map. Let M be an R-module and I an ideal of R. Given f ∈ M , the initial form of f in grI M , written in(f ) , is the equivalence class of f in I m M /I m+1 M where m is the maximum integer such that f ∈ I m M . If f ∈ I m M for every m, then set in(f ) = 0 . The initial form map is only a map of sets and generally not a homomorphism. For a submodule N ⊂ M , in(N ) is defined to be the submodule of grI M generated by {in(f )|f ∈ N } . This may not be the same as the submodule of grI M generated by the only initial forms of the generators of N. A ring inherits some “good” properties from its associated graded ring. For example, if R is a noetherian local ring, and grI R is an integral domain, then R is itself an integral domain.[1]

21.2 Examples Let U be the enveloping algebra of a Lie algebra g over a field k; it is filtered by degree. The Poincaré–Birkhoff–Witt theorem implies that gr U is a polynomial ring; in fact, it is the coordinate ring k[g∗ ] . The associated graded algebra of a Clifford algebra is an exterior algebra; i.e., a Clifford algebra degenerates to an exterior algebra.

21.3 Generalization to multiplicative filtrations The associated graded can also be defined more generally for multiplicative descending filtrations of R, Let F be a descending chain of ideals of the form 43

44

CHAPTER 21. ASSOCIATED GRADED RING

R = I0 ⊃ I1 ⊃ I2 ⊃ · · · such that Ij Ik ⊂ Ij+k . The graded ring associated with this filtration is grF R = ⊕∞ n=0 In /In+1 . Multiplication and the initial form map are defined as above.

21.4 See also • Graded (mathematics) • Rees algebra

21.5 References [1] Eisenbud, Corollary 5.5

• Eisenbud, David (1995). Commutative Algebra. Graduate Texts in Mathematics 150. New York: SpringerVerlag. doi:10.1007/978-1-4612-5350-1. ISBN 0-387-94268-8. MR 1322960. • Matsumura, Hideyuki (1989). Commutative ring theory. Cambridge Studies in Advanced Mathematics 8. Translated from the Japanese by M. Reid (Second ed.). Cambridge: Cambridge University Press. ISBN 0521-36764-6. MR 1011461.

Chapter 22

Associated prime In abstract algebra, an associated prime of a module M over a ring R is a type of prime ideal of R that arises as an annihilator of a (prime) submodule of M. The set of associated primes is usually denoted by AssR (M ) . In commutative algebra, associated primes are linked to the Lasker-Noether primary decomposition of ideals in commutative Noetherian rings. Specifically, if an ideal J is decomposed as a finite intersection of primary ideals, the radicals of these primary ideals are prime ideals, and this set of prime ideals coincides with AssR (R/J) .[1] Also linked with the concept of “associated primes” of the ideal are the notions of isolated primes and embedded primes.

22.1 Definitions A nonzero R module N is called a prime module if the annihilator AnnR (N ) = AnnR (N ′ ) for any nonzero submodule N' of N. For a prime module N, AnnR (N ) is a prime ideal in R.[2] An associated prime of an R module M is an ideal of the form AnnR (N ) where N is a prime submodule of M. In commutative algebra the usual definition is different, but equivalent:[3] if R is commutative, an associated prime P of M is a prime ideal of the form AnnR (m) for a nonzero element m of M or equivalently R/P is isomorphic to a submodule of M. In a commutative ring R, minimal elements in AssR (M ) (with respect to the set-theoretic inclusion) are called isolated primes while the rest of the associated primes (i.e., those properly containing associated primes) are called embedded prime. A module is called coprimary if xm = 0 for some nonzero m ∈ M implies xn M = 0 for some positive integer n. A nonzero finitely generated module M over a commutative Noetherian ring is coprimary if and only if it has exactly one associated prime. A submodule N of M is called P-primary if M /N is coprimary with P. An ideal I is a P-primary ideal if and only if AssR (R/I) = {P } ; thus, the notion is a generalization of a primary ideal.

22.2 Properties Most of these properties and assertions are given in (Lam 2001) starting on page 86. • If M' ⊆M, then AssR (M ′ ) ⊆ AssR (M ) . If in addition M' is an essential submodule of M, their associated primes coincide. • It is possible, even for a commutative local ring, that the set of associated primes of a finitely generated module is empty. However, in any ring satisfying the ascending chain condition on ideals (for example, any right or left Noetherian ring) every nonzero module has at least one associated prime. • Any uniform module has either zero or one associated primes, making uniform modules an example of coprimary modules. 45

46

CHAPTER 22. ASSOCIATED PRIME • For a one-sided Noetherian ring, there is a surjection from the set of isomorphism classes of indecomposable injective modules onto the spectrum Spec(R) . If R is an Artinian ring, then this map becomes a bijection. • Matlis’ Theorem: For a commutative Noetherian ring R, the map from the isomorphism classes of indecomposable injective modules to the spectrum is a bijection. Moreover, a complete set of representatives for those classes is given by E(R/p) where E(−) denotes the injective hull and p ranges over the prime ideals of R. • For a Noetherian module M over any ring, there are only finitely many associated primes of M.

The following properties all refer to a commutative Noetherian ring R: • Every ideal J (through primary decomposition) is expressible as a finite intersection of primary ideals. The radical of each of these ideals is a prime ideal, and these primes are exactly the elements of AssR (R/J) . In particular, an ideal J is a primary ideal if and only if AssR (R/J) has exactly one element. • Any prime ideal minimal with respect to containing an ideal J is in AssR (R/J) . These primes are precisely the isolated primes. • The set theoretic union of the associated primes of M is exactly the collection of zero-divisors on M, that is, elements r for which there exists nonzero m in M with mr =0. • If M is a finitely generated module over R, then there is a finite ascending sequence of submodules

0 = M0 ⊂ M1 ⊂ · · · ⊂ Mn−1 ⊂ Mn = M such that each quotient Mi/Mi−1 is isomorphic to R/Pi for some prime ideals Pi. Moreover every associated prime of M occurs among the set of primes Pi. (In general not all the ideals Pi are associated primes of M.) • Let S be a multiplicatively closed subset of R and f : Spec(S −1 R) → Spec(R) the canonical map. Then, for a module M over R, AssR (S −1 M ) = f (AssS −1 R (S −1 M )) = AssR (M ) ∩ {P |P ∩ S = ∅} .[4] • For a module M over R, Ass(M ) ⊆ Supp(M ) . Furthermore, the set of minimal elements of Supp(M ) coincides with the set of minimal elements of Ass(M ) . In particular, the equality holds if Ass(M ) consists of maximal ideals. • A module M over R has finite length if and only if M is finitely generated and Ass(M ) consists of maximal ideals.[5]

22.3 Examples • If R is the ring of integers, then non-trivial free abelian groups and non-trivial abelian groups of prime power order are coprimary. • If R is the ring of integers and M a finite abelian group, then the associated primes of M are exactly the primes dividing the order of M. • The group of order 2 is a quotient of the integers Z (considered as a free module over itself), but its associated prime ideal (2) is not an associated prime of Z.

22.4. REFERENCES

47

22.4 References [1] Lam 1999, p. 117, Ex 40B. [2] Lam 1999, p. 85. [3] Lam 1999, p. 86. [4] Matsumura 1970, 7.C Lemma [5] Cohn, P. M. (2003), Basic Algebra, Springer, Exercise 10.9.7, p. 391, ISBN 9780857294289.

• Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics 150, Berlin, New York: SpringerVerlag, ISBN 978-0-387-94268-1, MR 1322960 • Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294 • Matsumura, Hideyuki (1970), Commutative algebra

Chapter 23

Augmentation ideal In algebra, an augmentation ideal is an ideal that can be defined in any group ring. If G is a group and R a commutative ring, there is a ring homomorphism ε , called the augmentation map, from the group ring

R[G] to R, defined by taking a sum ∑

ri gi

to ∑

ri .

Here ri is an element of R and gi an element of G. The sums are finite, by definition of the group ring. In less formal terms,

ε(g) is defined as 1R whatever the element g in G, and ε is then extended to a homomorphism of R-modules in the obvious way. The augmentation ideal is the kernel of ε , and is therefore a two-sided ideal in R[G]. It is generated by the differences

g − g′ of group elements. Furthermore it is also generated by

g − 1, g ∈ G which is a basis for the augmentation ideal as a free R module. For R and G as above, the group ring R[G] is an example of an augmented R-algebra. Such an algebra comes equipped with a ring homomorphism to R. The kernel of this homomorphism is the augmentation ideal of the algebra. Another class of examples of augmentation ideal can be the kernel of the counit ε of any Hopf algebra. The augmentation ideal plays a basic role in group cohomology, amongst other applications. 48

23.1. REFERENCES

49

23.1 References • D. L. Johnson (1990). Presentations of groups. London Mathematical Society Student Texts 15. Cambridge University Press. pp. 149–150. ISBN 0-521-37203-8. • Dummit and Foote, Abstract Algebra

Chapter 24

Azumaya algebra In mathematics, an Azumaya algebra is a generalization of central simple algebras to R-algebras where R need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where R is a commutative local ring. The notion was developed further in ring theory, and in algebraic geometry, where Alexander Grothendieck made it the basis for his geometric theory of the Brauer group in Bourbaki seminars from 1964-5. There are now several points of access to the basic definitions. An Azumaya algebra over a commutative local ring R is an R-algebra A that is free and of finite rank r≥1 as an Rmodule, such that the tensor product A ⊗R A◦ (where Ao is the opposite algebra) is isomorphic to the matrix algebra EndR(A) ≈ Mr(R) via the map sending a ⊗ b to the endomorphism x → axb of A. An Azumaya algebra on a scheme X with structure sheaf OX, according to the original Grothendieck seminar, is a sheaf A of OX-algebras that is étale locally isomorphic to a matrix algebra sheaf; one should, however, add the condition that each matrix algebra sheaf is of positive rank. Milne, Étale Cohomology, starts instead from the definition that it is a sheaf A of OX-algebras whose stalk Ax at each point x is an Azumaya algebra over the local ring OX,x in the sense given above. Two Azumaya algebras A1 and A2 are equivalent if there exist locally free sheaves E 1 and E 2 of finite positive rank at every point such that

A1 ⊗ End(E1 ) ≃ A2 ⊗ End(E2 ), where End(Eᵢ) is the endomorphism sheaf of Ei. The Brauer group of X (an analogue of the Brauer group of a field) is the set of equivalence classes of Azumaya algebras. The group operation is given by tensor product, and the inverse is given by the opposite algebra. There have been significant applications of Azumaya algebras in diophantine geometry, following work of Yuri Manin. The Manin obstruction to the Hasse principle is defined using the Brauer group of schemes.

24.1 References • Knus, Max-Albert (1991), Quadratic and Hermitian forms over rings, Grundlehren der Mathematischen Wissenschaften 294, Berlin etc.: Springer-Verlag, ISBN 3-540-52117-8, Zbl 0756.11008 • Knus, Max-Albert; Ojanguren, Manuel (1974), Théorie de la descente et algèbres d'Azumaya, Lecture Notes in Mathematics 389, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0057799, MR 0417149, Zbl 0284.13002 • Saltman, David J. (1999). Lectures on division algebras. Regional Conference Series in Mathematics 94. Providence, RI: American Mathematical Society. ISBN 0-8218-0979-2. Zbl 0934.16013.

50

Chapter 25

Baer ring In abstract algebra and functional analysis, Baer rings, Baer *-rings, Rickart rings, Rickart *-rings, and AW*algebras are various attempts to give an algebraic analogue of von Neumann algebras, using axioms about annihilators of various sets. Any von Neumann algebra is a Baer *-ring, and much of the theory of projections in von Neumann algebras can be extended to all Baer *-rings, For example, Baer *-rings can be divided into types I, II, and III in the same way as von Neumann algebras. In the literature, left Rickart rings have also been termed left PP-rings. (“Principal implies projective": See definitions below.)

25.1 Definitions • An idempotent element of a ring is an element e which has the property that e2 = e. • The left annihilator of a set X ⊆ R is {r ∈ R | rX = {0}} • A (left) Rickart ring is a ring satisfying any of the following conditions: 1. the left annihilator of any single element of R is generated (as a left ideal) by an idempotent element. 2. (For unital rings) the left annihilator of any element is a direct summand of R. 3. All principal left ideals (ideals of the form Rx) are projective R modules.[1] • A Baer ring has the following definitions: 1. The left annihilator of any subset of R is generated (as a left ideal) by an idempotent element. 2. (For unital rings) The left annihilator of any subset of R is a direct summand of R.[2] For unital rings, replacing all occurrences of 'left' with 'right' yields an equivalent definition, that is to say, the definition is left-right symmetric.[3] In operator theory, the definitions are strengthened slightly by requiring the ring R to have an involution ∗ : R → R . Since this makes R isomorphic to its opposite ring Rop , the definition of Rickart *-ring is left-right symmetric. • A projection in a *-ring is an idempotent p that is self adjoint (p*=p). • A Rickart *-ring is a *-ring such that left annihilator of any element is generated (as a left ideal) by a projection. • A Baer *-ring is a *-ring such that left annihilator of any subset is generated (as a left ideal) by a projection. • An AW* algebra, introduced by Kaplansky (1951), is a C* algebra that is also a Baer *-ring. 51

52

CHAPTER 25. BAER RING

25.2 Examples • Since the principal left ideals of a left hereditary ring or left semihereditary ring are projective, it is clear that both types are left Rickart rings. This includes von Neumann regular rings, which are left and right semihereditary. If a von Neumann regular ring R is also right or left self injective, then R is Baer. • Any semisimple ring is Baer, since all left and right ideals are summands in R, including the annihilators. • Any domain is Baer, since all annihilators are {0} except for the annihilator of 0, which is R, and both {0} and R are summands of R. • The ring of bounded linear operators on a Hilbert space are a Baer ring and is also a Baer *-ring with the involution * given by the adjoint. • von Neumann algebras are examples of all the different sorts of ring above.

25.3 Properties The projections in a Rickart *-ring form a lattice, which is complete if the ring is a Baer *-ring.

25.4 See also • Baer *-semigroup

25.5 Notes [1] Rickart rings are named after Rickart (1946) who studied a similar property in operator algebras. This “principal implies projective” condition is the reason Rickart rings are sometimes called PP-rings. (Lam 1999) [2] This condition was studied by Reinhold Baer (1952). [3] T.Y. Lam (1999), “Lectures on Modules and Rings” ISBN 0-387-98428-3 pp.260

25.6 References • Baer, Reinhold (1952), Linear algebra and projective geometry, Boston, MA: Academic Press, ISBN 978-0486-44565-6, MR 0052795 • Berberian, Sterling K. (1972), Baer *-rings, Die Grundlehren der mathematischen Wissenschaften 195, Berlin, New York: Springer-Verlag, ISBN 978-3-540-05751-2, MR 0429975 • Kaplansky, Irving (1951), “Projections in Banach algebras”, Annals of Mathematics. Second Series 53 (2): 235–249, doi:10.2307/1969540, ISSN 0003-486X, JSTOR 1969540, MR 0042067 • Kaplansky, I. (1968), Rings of Operators, New York: W. A. Benjamin, Inc. • Rickart, C. E. (1946), “Banach algebras with an adjoint operation”, Annals of Mathematics. Second Series 47 (3): 528–550, doi:10.2307/1969091, JSTOR 1969091, MR 0017474 • L.A. Skornyakov (2001), “Regular ring (in the sense of von Neumann)", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • L.A. Skornyakov (2001), “Rickart ring”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • J.D.M. Wright (2001), “AW* algebra”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 26

Balanced module In the subfield of abstract algebra known as module theory, a right R module M is called a balanced module (or is said to have the double centralizer property) if every endomorphism of the abelian group M which commutes with all R-endomorphisms of M is given by multiplication by a ring element. Explicitly, for any additive endomorphism f, if fg = gf for every R endomorphism g, then there exists an r in R such that f(x) = xr for all x in M. In the case of non-balanced modules, there will be such an f that is not expressible this way. In the language of centralizers, a balanced module is one satisfying the conclusion of the double centralizer theorem, that is, the only endomorphisms of the group M commuting with all the R endomorphisms of M are the ones induced by right multiplication by ring elements. A ring is called balanced if every right R module is balanced.[1] It turns out that being balanced is a left-right symmetric condition on rings, and so there is no need to prefix it with “left” or “right”. The study of balanced modules and rings is an outgrowth of the study of QF-1 rings by C.J. Nesbitt and R. M. Thrall. This study was continued in V. P. Camillo's dissertation, and later it became fully developed. The paper (Dlab & Ringel 1972) gives a particularly broad view with many examples. In addition to these references, K. Morita and H. Tachikawa have also contributed published and unpublished results. A partial list of authors contributing to the theory of balanced modules and rings can be found in the references.

26.1 Examples and properties Examples • Every nonzero right ideal over a simple ring is balanced.[2] • Every faithful module over a quasi-Frobenius ring is balanced.[3] • The double centralizer theorem for right Artinian rings states that any simple right R module is balanced. • The paper (Dlab & Ringel 1972) contains numerous constructions of nonbalanced modules. • It was established in (Nesbitt & Thrall 1946) that uniserial rings are balanced. Conversely, a balanced ring which is finitely generated as a module over its center is uniserial.[4] • Among commutative Artinian rings, the balanced rings are exactly the quasi-Frobenius rings.[5] Properties • Being “balanced” is a categorical property for modules, that is, it is preserved by Morita equivalence. Explicitly, if F(–) is a Morita equivalence from the category of R modules to the category of S modules, and if M is balanced, then F(M) is balanced. • The structure of balanced rings is also completely determined in (Dlab & Ringel 1972), and is outlined in (Faith 1999, pp. 222–224). 53

54

CHAPTER 26. BALANCED MODULE • In view of the last point, the property of being a balanced ring is a Morita invariant property. • The question of which rings have all finitely generated right R modules balanced has already been answered. This condition turns out to be equivalent to the ring R being balanced.[6]

26.2 Notes [1] The definitions of balanced rings and modules appear in (Camillo 1970), (Cunningham & Rutter 1972), (Dlab & Ringel 1972), and (Faith 1999). [2] Lam 2001, p.37. [3] Camillo & Fuller 1972. [4] Faith 1999, p.223. [5] Camillo 1970, Theorem 21. [6] Dlab & Ringel 1972.

26.3 References • Camillo, Victor P. (1970), “Balanced rings and a problem of Thrall”, Trans. Amer. Math. Soc. 149: 143–153, doi:10.1090/s0002-9947-1970-0260794-0, ISSN 0002-9947, MR 0260794 • Camillo, V. P.; Fuller, K. R. (1972), “Balanced and QF-1 algebras”, Proc. Amer. Math. Soc. 34: 373–378, doi:10.1090/s0002-9939-1972-0306256-0, ISSN 0002-9939, MR 0306256 • Cunningham, R. S.; Rutter, E. A., Jr. (1972), “The double centralizer property is categorical”, Rocky Mountain J. Math. 2 (4): 627–629, doi:10.1216/rmj-1972-2-4-627, ISSN 0035-7596, MR 0310017 • Dlab, Vlastimil; Ringel, Claus Michael (1972), “Rings with the double centralizer property”, J. Algebra 22: 480–501, doi:10.1016/0021-8693(72)90163-9, ISSN 0021-8693, MR 0306258 • Faith, Carl (1999), Rings and things and a fine array of twentieth century associative algebra, Mathematical Surveys and Monographs 65, Providence, RI: American Mathematical Society, pp. xxxiv+422, ISBN 0-82180993-8, MR 1657671 • Lam, T. Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics 131 (2 ed.), New York: Springer-Verlag, pp. xx+385, ISBN 0-387-95183-0, MR 1838439 • Nesbitt, C. J.; Thrall, R. M. (1946), “Some ring theorems with applications to modular representations”, Ann. of Math. (2) 47: 551–567, doi:10.2307/1969092, ISSN 0003-486X, MR 0016760

Chapter 27

Beauville–Laszlo theorem In mathematics, the Beauville–Laszlo theorem is a result in commutative algebra and algebraic geometry that allows one to “glue” two sheaves over an infinitesimal neighborhood of a point on an algebraic curve. It was proved by Arnaud Beauville and Yves Laszlo (1995).

27.1 The theorem Although it has implications in algebraic geometry, the theorem is a local result and is stated in its most primitive form for commutative rings. If A is a ring and f is a nonzero element of A, then we can form two derived rings: the localization at f, Af, and the completion at Af, Â; both are A-algebras. In the following we assume that f is a non-zero divisor. Geometrically, A is viewed as a scheme X = Spec A and f as a divisor (f) on Spec A; then Af is its complement Df = Spec Af, the principal open set determined by f, while  is an “infinitesimal neighborhood” D = Spec  of (f). The intersection of Df and Spec  is a “punctured infinitesimal neighborhood” D0 about (f), equal to Spec  ⊗A Af = Spec Âf. Suppose now that we have an A-module M; geometrically, M is a sheaf on Spec A, and we can restrict it to both the principal open set Df and the infinitesimal neighborhood Spec Â, yielding an Af-module F and an Â-module G. Algebraically,

F = M ⊗A Af = Mf

ˆ G = M ⊗A A.

c , meaning the completion of the A-module M at the ideal Af, (Despite the notational temptation to write G = M unless A is noetherian and M is finitely-generated, the two are not in fact equal. This phenomenon is the main reason that the theorem bears the names of Beauville and Laszlo; in the noetherian, finitely-generated case, it is, as noted by the authors, a special case of Grothendieck’s faithfully flat descent.) F and G can both be further restricted to the punctured neighborhood D0 , and since both restrictions are ultimately derived from M, they are isomorphic: we have an isomorphism ∼ ˆ ϕ : Gf − →F ⊗Af Aˆf = F ⊗A A.

Now consider the converse situation: we have a ring A and an element f, and two modules: an Af-module F and an Â-module G, together with an isomorphism φ as above. Geometrically, we are given a scheme X and both an open set Df and a “small” neighborhood D of its closed complement (f); on Df and D we are given two sheaves which agree on the intersection D0 = Df ∩ D. If D were an open set in the Zariski topology we could glue the sheaves; the content of the Beauville–Laszlo theorem is that, under one technical assumption on f, the same is true for the infinitesimal neighborhood D as well. Theorem: Given A, f, F, G, and φ as above, if G has no f-torsion, then there exist an A-module M and isomorphisms ∼

α : Mf − →F

∼ β : M ⊗A Aˆ− →G

55

56

CHAPTER 27. BEAUVILLE–LASZLO THEOREM

consistent with the isomorphism φ: φ is equal to the composition

β −1 ⊗1

α⊗1

ˆ Gf = G ⊗A Af −−−−→M ⊗A Aˆ ⊗A Af = Mf ⊗A Aˆ−−−→F ⊗A A. The technical condition that G has no f-torsion is referred to by the authors as "f-regularity”. In fact, one can state a stronger version of this theorem. Let M(A) be the category of A-modules (whose morphisms are A-module homomorphisms) and let Mf(A) be the full subcategory of f-regular modules. In this notation, we obtain a commutative diagram of categories (note Mf(Af) = M(Af)):

ˆ Mf (A) −→ Mf (A) ↓ ↓ M(Af ) −→ M(Aˆf ) in which the arrows are the base-change maps; for example, the top horizontal arrow acts on objects by M → M ⊗A Â. Theorem: The above diagram is a cartesian diagram of categories.

27.2 Global version In geometric language, the Beauville–Laszlo theorem allows one to glue sheaves on a one-dimensional affine scheme over an infinitesimal neighborhood of a point. Since sheaves have a “local character” and since any scheme is locally affine, the theorem admits a global statement of the same nature. The version of this statement that the authors found noteworthy concerns vector bundles: Theorem: Let X be an algebraic curve over a field k, x a k-rational smooth point on X with infinitesimal neighborhood D = Spec k[[t]], R a k-algebra, and r a positive integer. Then the category Vectr(XR) of rank-r vector bundles on the curve XR = X ×S ₑ k Spec R fits into a cartesian diagram:

Vectr (XR ) ↓ Vectr ((X \ x)R )

−→ Vectr (DR ) ↓ 0 −→ Vectr (DR )

This entails a corollary stated in the paper: Corollary: With the same setup, denote by Triv(XR) the set of triples (E, τ, σ), where E is a vector bundle on XR, τ is a trivialization of E over (X \ x)R (i.e., an isomorphism with the trivial bundle O₍X - x₎R), and σ a trivialization over DR. Then the maps in the above diagram furnish a bijection between Triv(XR) and GLr(R((t))) (where R((t)) is the formal Laurent series ring). The corollary follows from the theorem in that the triple is associated with the unique matrix which, viewed as a “transition function” over D0 R between the trivial bundles over (X \ x)R and over DR, allows gluing them to form E, with the natural trivializations of the glued bundle then being identified with σ and τ. The importance of this corollary is that it shows that the affine Grassmannian may be formed either from the data of bundles over an infinitesimal disk, or bundles on an entire algebraic curve.

27.3 References • Beauville, Arnaud; Laszlo, Yves (1995), “Un lemme de descente”, Comptes Rendus de l'Académie des Sciences. Série I. Mathématique 320 (3): 335–340, ISSN 0764-4442, retrieved 2008-04-08

Chapter 28

Bimodule In abstract algebra a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in the sense that many of the relationships between left and right modules become simpler when they are expressed in terms of bimodules.

28.1 Definition If R and S are two rings, then an R-S-bimodule is an abelian group M such that: 1. M is a left R-module and a right S-module. 2. For all r in R, s in S and m in M:

(rm)s = r(ms). An R-R-bimodule is also known as an R-bimodule.

28.2 Examples • For positive integers n and m, the set Mn,m(R) of n × m matrices of real numbers is an R-S bimodule, where R is the ring Mn(R) of n × n matrices, and S is the ring Mm(R) of m × m matrices. Addition and multiplication are carried out using the usual rules of matrix addition and matrix multiplication; the heights and widths of the matrices have been chosen so that multiplication is defined. Note that Mn,m(R) itself is not a ring (unless n = m), because multiplying an n × m matrix by another n × m matrix is not defined. The crucial bimodule property, that (rx)s = r(xs), is the statement that multiplication of matrices is associative. • If R is a ring, then R itself is an R-bimodule, and so is Rn (the n-fold direct product of R). • Any two-sided ideal of a ring R is an R-bimodule. • Any module over a commutative ring R is automatically a bimodule. For example, if M is a left module, we can define multiplication on the right to be the same as multiplication on the left. (However, not all R-bimodules arise this way.) • If M is a left R-module, then M is an R-Z bimodule, where Z is the ring of integers. Similarly, right R-modules may be interpreted as Z-R bimodules, and indeed an abelian group may be treated as a Z-Z bimodule. • If R is a subring of S, then S is an R-bimodule. It is also an R-S and an S-R bimodule. 57

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CHAPTER 28. BIMODULE

28.3 Further notions and facts If M and N are R-S bimodules, then a map f : M → N is a bimodule homomorphism if it is both a homomorphism of left R-modules and of right S-modules. An R-S bimodule is actually the same thing as a left module over the ring R ⊗Z S op , where S op is the opposite ring of S (with the multiplication turned around). Bimodule homomorphisms are the same as homomorphisms of left R ⊗Z S op modules. Using these facts, many definitions and statements about modules can be immediately translated into definitions and statements about bimodules. For example, the category of all R-S bimodules is abelian, and the standard isomorphism theorems are valid for bimodules. There are however some new effects in the world of bimodules, especially when it comes to the tensor product: if M is an R-S bimodule and N is an S-T bimodule, then the tensor product of M and N (taken over the ring S) is an R-T bimodule in a natural fashion. This tensor product of bimodules is associative (up to a unique canonical isomorphism), and one can hence construct a category whose objects are the rings and whose morphisms are the bimodules. This is in fact a 2-category, in a canonical way—2 morphisms between R-S bimodules M and N are exactly bimodule homomorphisms, i.e. functions

f :M →N satisfying 1. f (m + m′ ) = f (m) + f (m′ ) 2. f (rms) = rf (m)s , for m∈M, r∈R, and s∈S. One immediately verifies the interchange law for bimodule homomorphisms, i.e.

(f ′ ⊗ g ′ ) ◦ (f ⊗ g) = (f ′ ◦ f ) ⊗ (g ′ ◦ g) holds whenever either (and hence the other) side of the equation is defined, and where ∘ is the usual composition of homomorphisms. In this interpretation, the category End(R)=Bimod(R,R) is exactly the monoidal category of R-R bimodules with the usual tensor product over R the tensor product of the category. In particular, if R is a commutative ring, every left or right R-module is canonically an R-R bimodule, and the category Bimod(R,R)=RMod is symmetric monoidal. The case that R is a field K is a motivating example of a symmetric monoidal category, in which case R-Mod = K-Vect, the category of vector spaces over K, with the usual tensor product ⊗ = ⊗K giving the monoidal structure, and with unit K. We also see that a monoid in Bimod(R,R) is exactly an R-algebra. See (Street 2003).[1] Furthermore, if M is an R-S bimodule and L is an T-S bimodule, then the set HomS(M,L) of all S-module homomorphisms from M to L becomes a T-R module in a natural fashion. These statements extend to the derived functors Ext and Tor. Profunctors can be seen as a categorical generalization of bimodules. Note that bimodules are not at all related to bialgebras.

28.4 See also • profunctor

28.5 References [1] Street, Ross (20 Mar 2003). “Categorical and combinatorial aspects of descent theory”. Retrieved 11 August 2014.

• p133–136, Jacobson, N. (1989). Basic Algebra II. W. H. Freeman and Company. ISBN 0-7167-1933-9.

Chapter 29

Binomial ring In mathematics, a binomial ring is a ring whose additive group is torsion-free that contains all binomial coefficients ( ) x x(x − 1) · · · (x − n + 1) = n n! for x in the ring and n a positive integer. Binomial rings were introduced by Hall (1969). Elliott (2006) showed that binomial rings are essentially the same as λ-rings such that all Adams operations are the identity.

29.1 References • Elliott, Jesse (2006), “Binomial rings, integer-valued polynomials, and λ-rings”, Journal of Pure and Applied Algebra 207 (1): 165–185, doi:10.1016/j.jpaa.2005.09.003, ISSN 0022-4049, MR 2244389 • Hall, Philip (1969) [1957], The Edmonton notes on nilpotent groups. Notes of lectures given at the Canadian Mathematical Congress Summer Seminar (University of Alberta, 12–30 august 1957), Queen Mary College Mathematics Notes, Mathematics Department, Queen Mary College, London, ISBN 978-0-902480-06-3, MR 0283083 • Yau, Donald (2010), Lambda-rings, Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., ISBN 978981-4299-09-1, MR 2649360

59

Chapter 30

Biquaternion In abstract algebra, the biquaternions are the numbers w + x i + y j + z k, where w, x, y, and z are complex numbers and the elements of {1, i, j, k} multiply as in the quaternion group. As there are three types of complex number, so there are three types of biquaternion: • (Ordinary) biquaternions when the coefficients are (ordinary) complex numbers • Split-biquaternions when w, x, y, and z are split-complex numbers • Dual quaternions when w, x, y, and z are dual numbers. This article is about the ordinary biquaternions named by William Rowan Hamilton in 1844 (see Proceedings of Royal Irish Academy 1844 & 1850 page 388). Some of the more prominent proponents of these biquaternions include Alexander Macfarlane, Arthur W. Conway, Ludwik Silberstein, and Cornelius Lanczos. As developed below, the unit quasi-sphere of the biquaternions provides a presentation of the Lorentz group, which is the foundation of special relativity. The algebra of biquaternions can be considered as a tensor product C ⊗ H (taken over the reals) where C is the field of complex numbers and H is the algebra of (real) quaternions. In other words, the biquaternions are just the complexification of the (real) quaternions. Viewed as a complex algebra, the biquaternions are isomorphic to the algebra of 2×2 complex matrices M 2 (C). They can be classified as the Clifford algebra Cℓ2 (C) = Cℓ0 3 (C). This is also isomorphic to the Pauli algebra Cℓ₃,₀(R), and the even part of the spacetime algebra Cℓ0 ₁,₃(R).

30.1 Definition Let {1, i, j, k} be the basis for the (real) quaternions, and let u, v, w, x be complex numbers, then q=u1+vi+wj+xk is a biquaternion.[1] To distinguish square roots of minus one in the biquaternions, Hamilton[2][3] and Arthur W. Conway used the convention of representing the square root of minus one in the scalar field C by h since there is an i in the quaternion group. Then h i = i h, h j = j h, and h k = k h since h is a scalar. Hamilton’s primary exposition on biquaternions came in 1853 in his Lectures on Quaternions, now available in the Historical Mathematical Monographs of Cornell University. The two editions of Elements of Quaternions (1866 & 1899) reduced the biquaternion coverage in favor of the real quaternions. He introduced the terms bivector, biconjugate, bitensor, and biversor. Considered with the operations of component-wise addition, and multiplication according to the quaternion group, this collection forms a 4-dimensional algebra over the complex numbers. The algebra of biquaternions is associative, but not commutative. A biquaternion is either a unit or a zero divisor. 60

30.2. PLACE IN RING THEORY

61

30.2 Place in ring theory 30.2.1

Linear representation

Note the matrix product ( h 0

)( ) ( ) 0 1 0 h 0 = −h −1 0 h 0

where each of these three arrays has a square equal to the negative of the identity matrix. When this matrix product is interpreted as i j = k, then one obtains a subgroup of the matrix group that is isomorphic to the quaternion group. Consequently (

u + hv w + hx −w + hx u − hv

)

represents biquaternion q = u 1 + v i + w j + x k. Given any 2 × 2 complex matrix, there are complex values u, v, w, and x to put it in this form so that the matrix ring is isomorphic[4] to the biquaternion ring.

30.2.2

Subalgebras

Considering the biquaternion algebra over the scalar field of real numbers R, the set {1, h, i, hi, j, hj, k, hk } forms a basis so the algebra has eight real dimensions. Note the squares of the elements hi, hj, and hk are all plus one, for example,

(hi)2 = h2 i2 = (−1)(−1) = +1. Then the subalgebra given by {x+y(hi) : x, y ∈ R} is ring isomorphic to the plane of split-complex numbers, which has an algebraic structure built upon the unit hyperbola. The elements hj and hk also determine such subalgebras. Furthermore, {x + yj : x, y ∈ C} is a subalgebra isomorphic to the tessarines. A third subalgebra called coquaternions is generated by hj and hk. First note that (hj)(hk) = (−1) i, and that the square of this element is −1. These elements generate the dihedral group of the square. The linear subspace with basis {1, i, hj, hk} thus is closed under multiplication, and forms the coquaternion algebra. In the context of quantum mechanics and spinor algebra, the biquaternions hi, hj, and hk (or their negatives), viewed in the M(2,C) representation, are called Pauli matrices.

30.3 Algebraic properties The biquaternions have two conjugations: • the biconjugate or biscalar minus bivector is q ∗ = w − xi − yj − zk , and • the complex conjugation of biquaternion coefficients q ⋆ = w⋆ + x⋆ i + y ⋆ j + z ⋆ k where z ⋆ = a − bh when z = a + bh, Note that (pq)∗ = q ∗ p∗ ,

a, b ∈ R,

(pq)⋆ = p⋆ q ⋆ ,

h2 = −1.

(q ∗ )⋆ = (q ⋆ )∗ .

Clearly, if qq ∗ = 0 then q is a zero divisor. Otherwise {qq ∗ }−1 is defined over the complex numbers. Further, qq ∗ = q ∗ q is easily verified. This allows an inverse to be defined as follows: • q −1 = q ∗ {qq ∗ }−1 , iff qq ∗ ̸= 0.

62

30.3.1

CHAPTER 30. BIQUATERNION

Relation to Lorentz transformations

Consider now the linear subspace [5]

M = {q : q ∗ = q ⋆ } = {t + x(hi) + y(hj) + z(hk) : t, x, y, z ∈ R}. M is not a subalgebra since it is not closed under products; for example (hi)(hj) = h2 ij = −k ∈ / M. . Indeed, M cannot form an algebra if it is not even a magma. Proposition: If q is in M, then qq ∗ = t2 − x2 − y 2 − z 2 . proof: qq ∗ = (t + xhi + yhj + zhk)(t − xhi − yhj − zhk)

= t2 − x2 (hi)2 − y 2 (hj)2 − z 2 (hk)2 = t2 − x2 − y 2 − z 2 . Definition: Let biquaternion g satisfy g g * = 1. Then the Lorentz transformation associated with g is given by

T (q) = g ∗ qg ⋆ . Proposition: If q is in M, then T(q) is also in M. proof: (g ∗ qg ⋆ )∗ = (g ⋆ )∗ q ∗ g = (g ∗ )⋆ q ⋆ g = (g ∗ qg ⋆ )⋆ . Proposition:

T (q)(T (q))∗ = qq ∗

proof: Note first that g g * = 1 means that the sum of the squares of its four complex components is one. Then the sum of the squares of the complex conjugates of these components is also one. Therefore, g ⋆ (g ⋆ )∗ = 1. Now

(g ∗ qg ⋆ )(g ∗ qg ⋆ )∗ = g ∗ qg ⋆ (g ⋆ )∗ q ∗ g = g ∗ qq ∗ g = qq ∗ .

30.4 Associated terminology As the biquaternions have been a fixture of linear algebra since the beginnings of mathematical physics, there is an array of concepts that are illustrated or represented by biquaternion algebra. The transformation group G = {g : gg ∗ = 1} has two parts, G ∩ H and G ∩ M. The first part is characterized by g = g ⋆ ; then the Lorentz transformation corresponding to g is given by T (q) = g −1 qg since g ∗ = g −1 . Such a transformation is a rotation by quaternion multiplication, and the collection of them is O(3) ∼ = G ∩ H. But this subgroup of G is not a normal subgroup, so no quotient group can be formed. To view G ∩ M it is necessary to show some subalgebra structure in the biquaternions. Let r represent an element of the sphere of square roots of minus one in the real quaternion subalgebra H. Then (hr)2 = +1 and the plane of biquaternions given by Dr = {z = x + yhr : x, y ∈ R} is a commutative subalgebra isomorphic to the plane of split-complex numbers. Just as the ordinary complex plane has a unit circle, Dr has a unit hyperbola given by

exp(ahr) = cosh(a) + hr sinh(a),

a ∈ R.

Just as the unit circle turns by multiplication through one of its elements, so the hyperbola turns because exp(ahr) exp(bhr) = exp((a + b)hr).Hence these algebraic operators on the hyperbola are called hyperbolic versors. The unit circle in C and unit hyperbola in Dr are examples of one-parameter groups. For every square root r of minus one in H, there is a one-parameter group in the biquaternions given by G ∩ Dr . The space of biquaternions has a natural topology through the Euclidean metric on 8-space. With respect to this topology, G is a topological group. Moreover, it has analytic structure making it a six-parameter Lie group. Consider the subspace of bivectors A = {q : q ∗ = −q} . Then the exponential map exp : A → G takes the real vectors to G ∩ H and the h-vectors to G ∩ M. When equipped with the commutator, A forms the Lie algebra of G. Thus this

30.5. SEE ALSO

63

study of a six-dimensional space serves to introduce the general concepts of Lie theory. When viewed in the matrix representation, G is called the special linear group SL(2,C) in M(2,C). Many of the concepts of special relativity are illustrated through the biquaternion structures laid out. The subspace M corresponds to Minkowski space, with the four coordinates giving the time and space locations of events in a resting frame of reference. Any hyperbolic versor exp(ahr) corresponds to a velocity in direction r of speed c tanh a where c is the velocity of light. The inertial frame of reference of this velocity can be made the resting frame by applying the Lorentz boost T given by g = exp(0.5ahr) since then g ⋆ = exp(−0.5ahr) = g ∗ so that T (exp(ahr)) = 1. Naturally the hyperboloid G ∩ M, which represents the range of velocities for sub-luminal motion, is of physical interest. There has been considerable work associating this “velocity space” with the hyperboloid model of hyperbolic geometry. In special relativity, the hyperbolic angle parameter of a hyperbolic versor is called rapidity. Thus we see the biquaternion group G provides a group representation for the Lorentz group. After the introduction of spinor theory, particularly in the hands of Wolfgang Pauli and Élie Cartan, the biquaternion representation of the Lorentz group was superseded. The new methods were founded on basis vectors in the set

{q : qq ∗ = 0} = {w + xi + yj + zk : w2 + x2 + y 2 + z 2 = 0} which is called the “complex light cone”.

30.5 See also • Biquaternion algebra • Conic octonions (isomorphism) • MacFarlane’s use • Quotient ring • Hypercomplex number

30.6 Notes [1] Hamilton (1853) page 639 [2] Hamilton (1853) page 730 [3] Hamilton (1899) Elements of Quaternions, 2nd edition, page 289 [4] Leonard Dickson (1914) Linear Algebras, §13 “Equivalence of the complex quaternion and matric algebras”, page 13 [5] Lanczos (1949) Equation 94.16 page 305. The following algebra compares to Lanczos, except he uses ~ to signify quaternion conjugation and * for complex conjugation

30.7 References • Proceedings of the Royal Irish academy November 1844 (NA) and 1850 page 388 from google books • Arthur Buchheim (1885) “A Memoir on biquaternions”, American Journal of Mathematics 7(4):293 to 326 from Jstor early content. • Conway, Arthur W. (1911), “On the application of quaternions to some recent developments in electrical theory”, Proceedings of the Royal Irish Academy 29A: 1–9. • William Rowan Hamilton (1853) Lectures on Quaternions, Article 669. This historical mathematical text is available on-line courtesy of Cornell University.

64

CHAPTER 30. BIQUATERNION • Hamilton (1866) Elements of Quaternions University of Dublin Press. Edited by William Edwin Hamilton, son of the deceased author. • Hamilton (1899) Elements of Quaternions volume I, (1901) volume II. Edited by Charles Jasper Joly; published by Longmans, Green & Co.. • Kravchenko, Vladislav (2003), Applied Quaternionic Analysis, Heldermann Verlag ISBN 3-88538-228-8. • Lanczos, Cornelius (1949), The Variational Principles of Mechanics, University of Toronto Press, pp. 304–312. • Silberstein, Ludwik (1912), “Quaternionic form of relativity”, Philosophical Magazine, Series 6 23: 790–809, doi:10.1080/14786440508637276. • Silberstein, Ludwik (1914), The Theory of Relativity. • Synge, J. L. (1972), “Quaternions, Lorentz transformations, and the Conway-Dirac-Eddington matrices”, Communications of the Dublin Institute for Advanced Studies, Series A 21. • Girard, P. R. (1984), “The quaternion group and modern physics”, European Journal of Physics 5: 25–32, Bibcode:1984EJPh....5...25G, doi:10.1088/0143-0807/5/1/007. • Kilmister, C. W. (1994), Eddington’s search for a fundamental theory, Cambridge University Press, pp. 121, 122, 179, 180, ISBN 0-521-37165-1. • Sangwine, Stephen J.; Ell, Todd A.; Le Bihan, Nicolas (2010), “Fundamental representations and algebraic properties of biquaternions or complexified quaternions”, Advances in Applied Clifford Algebras: 1–30, arXiv:1001.0240, doi:10.1007/s00006-010-0263-3. • Sangwine, Stephen J.; Alfsmann, Daniel (2010), “Determination of the biquaternion divisors of zero, including idempotents and nilpotents”, Advances in Applied Clifford Algebras 20 (2): 401–410, arXiv:0812.1102, Bibcode:2008arXiv0812.1102S, doi:10.1007/s00006-010-0202-3. • Tanişli, M. (2006), “Gauge transformation and electromagnetism with biquaternions”, Europhysics Letters 74 (4): 569, Bibcode:2006EL.....74..569T, doi:10.1209/epl/i2005-10571-6.

Chapter 31

Bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered an order zero quantity, and a vector is an order one quantity, then a bivector can be thought of as being of order two. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and quaternions in three dimensions. They can be used to generate rotations in any dimension, and are a useful tool for classifying such rotations. They also are used in physics, tying together a number of otherwise unrelated quantities. Bivectors are generated by the exterior product on vectors: given two vectors a and b, their exterior product a ∧ b is a bivector, as is the sum of any bivectors. Not all bivectors can be generated as a single exterior product. More precisely, a bivector that can be expressed as an exterior product is called simple; in up to three dimensions all bivectors are simple, but in higher dimensions this is not the case.[1] The exterior product is antisymmetric, so b ∧ a is the negation of the bivector a ∧ b, producing the opposite orientation, and a ∧ a is the zero bivector. Geometrically, a simple bivector can be interpreted as an oriented plane segment, much as vectors can be thought of as directed line segments.[3] The bivector a ∧ b has a magnitude equal to the area of the parallelogram with edges a and b, has the attitude of the plane spanned by a and b, and has orientation being the sense of the rotation that would align a with b.[3][4]

31.1 History The bivector was first defined in 1844 by German mathematician Hermann Grassmann in exterior algebra as the result of the exterior product of two vectors. Around the same time in 1843 in Ireland William Rowan Hamilton discovered quaternions. It was not until English mathematician William Kingdon Clifford in 1888 added the geometric product to Grassmann’s algebra, incorporating the ideas of both Hamilton and Grassmann, and founded Clifford algebra, that the bivector as it is known today was fully understood. Around this time Josiah Willard Gibbs and Oliver Heaviside developed vector calculus, which included separate cross product and dot products that were derived from quaternion multiplication.[5][6][7] The success of vector calculus, and of the book Vector Analysis by Gibbs and Wilson, had the effect that the insights of Hamilton and Clifford were overlooked for a long time, since much of 20th century mathematics and physics was formulated in vector terms. Gibbs used vectors to fill the role of bivectors in three dimensions, and used “bivector” to describe an unrelated quantity, a use that has sometimes been copied.[8][9][10] Today the bivector is largely studied as a topic in geometric algebra, a Clifford algebra over real or complex vector spaces with a nondegenerate quadratic form. Its resurgence was led by David Hestenes who, along with others, applied geometric algebra to a range of new applications in physics.[11]

31.2 Derivation For this article the bivector will be considered only in real geometric algebras. This in practice is not much of a restriction, as all useful applications are drawn from such algebras. Also unless otherwise stated, all examples have a Euclidean metric and so a positive-definite quadratic form. 65

66

CHAPTER 31. BIVECTOR

Parallel plane segments with the same orientation and area corresponding to the same bivector a ∧ b.[2]

31.2.1

Geometric algebra and the geometric product

The bivector arises from the definition of the geometric product over a vector space. For vectors a, b and c, the geometric product on vectors is defined as follows: Associativity:

(ab)c = a(bc)

31.2. DERIVATION

67

Left and right distributivity:

a(b + c) = ab + ac (b + c)a = ba + ca Contraction:

a2 = Q(a) = ϵa |a|

2

Where Q is the quadratic form, |a| is the magnitude of a and ϵₐ is the signature of the vector. For a space with Euclidean metric ϵₐ is 1 so can be omitted, and the contraction condition becomes:

a2 = |a|

31.2.2

2

The interior product

From associativity a(ab) = a2 b, a scalar times b. When b is not parallel to and hence not a scalar multiple of a, ab cannot be a scalar. But

1 1 (ab + ba) = ((a + b)2 − a2 − b2 ) 2 2 is a sum of scalars and so a scalar. From the law of cosines on the triangle formed by the vectors its value is |a||b|cosθ, where θ is the angle between the vectors. It is therefore identical to the interior product between two vectors, and is written the same way,

a·b=

1 (ab + ba). 2

It is symmetric, scalar valued, and can be used to determine the angle between two vectors: in particular if a and b are orthogonal the product is zero.

31.2.3

The exterior product

Just as the interior product can be formulated as the symmetric part of the geometric product another quantity, the exterior product can be formulated as its antisymmetric part:

a∧b=

1 (ab − ba) 2

It is antisymmetric in a and b

b∧a=

1 1 (ba − ab) = − (ab − ba) = −a ∧ b 2 2

and by addition:

a·b+a∧b=

1 1 (ab + ba) + (ab − ba) = ab 2 2

That is, the geometric product is the sum of the symmetric interior product and antisymmetric exterior product.

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CHAPTER 31. BIVECTOR

To examine the nature of a ∧ b, consider the formula

(a · b)2 − (a ∧ b)2 = a2 b2 , which using the Pythagorean trigonometric identity gives the value of (a ∧ b)2 2

2

2

2

(a ∧ b)2 = (a · b)2 − a2 b2 = |a| |b| (cos2 θ − 1) = − |a| |b| sin2 θ With a negative square it cannot be a scalar or vector quantity, so it is a new sort of object, a bivector. It has magnitude |a| |b| |sinθ|, where θ is the angle between the vectors, and so is zero for parallel vectors. To distinguish them from vectors, bivectors are written here with bold capitals, for example:

A = a ∧ b = −b ∧ a , although other conventions are used, in particular as vectors and bivectors are both elements of the geometric algebra.

31.3 Properties 31.3.1

The space Λ2 Rn

The algebra generated by the geometric product is the geometric algebra over the vector space. For a Euclidean vector space it is written Gn or Cℓn(ℝ), where n is the dimension of the vector space ℝn . Cℓn is both a vector space and an algebra, generated by all the products between vectors in ℝn , so it contains all vectors and bivectors. More precisely as a vector space it contains the vectors and bivectors as linear subspaces, though not subalgebras. The space of all bivectors is written Λ2 ℝn .[12]

31.3.2

The even subalgebra

The subalgebra generated by the bivectors is the even subalgebra of the geometric algebra, written Cℓ + n . This algebra results from considering all products of scalars and bivectors generated by the geometric product. It has dimension 2n − 1 , and contains Λ2 ℝn as a linear subspace with dimension 1/2n(n − 1) (a triangular number). In two and three dimensions the even subalgebra contains only scalars and bivectors, and each is of particular interest. In two dimensions the even subalgebra is isomorphic to the complex numbers, ℂ, while in three it is isomorphic to the quaternions, ℍ. More generally the even subalgebra can be used to generate rotations in any dimension, and can be generated by bivectors in the algebra.

31.3.3

Magnitude

As noted in the previous section the magnitude of a simple bivector, that is one that is the exterior product of two vectors a and b, is |a||b|sin θ, where θ is the angle between the vectors. It is written |B|, where B is the bivector. For general bivectors the magnitude can be calculated by taking the norm of the bivector considered as a vector in the space Λ2 ℝn . If the magnitude is zero then all the bivector’s components are zero, and the bivector is the zero bivector which as an element of the geometric algebra equals the scalar zero.

31.3.4

Unit bivectors

A unit bivector is one with unit magnitude. It can be derived from any non-zero bivector by dividing the bivector by its magnitude, that is B . |B|

31.4. TWO DIMENSIONS

69

Of particular interest are the unit bivectors formed from the products of the standard basis. If ei and ej are distinct basis vectors then the product ei ∧ ej is a bivector. As the vectors are orthogonal this is just eiej, written eij, with unit magnitude as the vectors are unit vectors. The set of all such bivectors form a basis for Λ2 ℝn . For instance in four dimensions the basis for Λ2 ℝ4 is (e1 e2 , e1 e3 , e1 e4 , e2 e3 , e2 e4 , e3 e4 ) or (e12 , e13 , e14 , e23 , e24 , e34 ).[13]

31.3.5

Simple bivectors

The exterior product of two vectors is a bivector, but not all bivectors are exterior products of two vectors. For example in four dimensions the bivector

B = e1 ∧ e2 + e3 ∧ e4 = e1 e2 + e3 e4 = e12 + e34 cannot be written as the exterior product of two vectors. A bivector that can be written as the exterior product of two vectors is simple. In two and three dimensions all bivectors are simple, but not in four or more dimensions; in four dimensions every bivector is the sum of at most two exterior products. A bivector has a real square if and only if it is simple, and only simple bivectors can be represented geometrically by an oriented plane area.[1]

31.3.6

Product of two bivectors

The geometric product of two bivectors, A and B, is

AB = A · B + A × B + A ∧ B. The quantity A · B is the scalar valued interior product, while A ∧ B is the grade 4 exterior product that arises in four or more dimensions. The quantity A × B is the bivector valued commutator product, given by A × B = 12 (AB − BA), [14] The space of bivectors Λ2 ℝn are a Lie algebra over ℝ, with the commutator product as the Lie bracket. The full geometric product of bivectors generates the even subalgebra. Of particular interest is the product of a bivector with itself. As the commutator product is antisymmetric the product simplifies to

AA = A · A + A ∧ A. If the bivector is simple the last term is zero and the product is the scalar valued A · A, which can be used as a check for simplicity. In particular the exterior product of bivectors only exists in four or more dimensions, so all bivectors in two and three dimensions are simple.[1]

31.4 Two dimensions When working with coordinates in geometric algebra it is usual to write the basis vectors as (e1 , e2 , ...), a convention that will be used here. A vector in real two-dimensional space ℝ2 can be written a = a1 e1 + a2 e2 , where a1 and a2 are real numbers, e1 and e2 are orthonormal basis vectors. The geometric product of two such vectors is

ab = (a1 e1 + a2 e2 )(b1 e1 + b2 e2 ) = a1 b1 e1 e1 + a1 b2 e1 e2 + a2 b1 e2 e1 + a2 b2 e2 e2 = a1 b1 + a2 b2 + (a1 b2 − a2 b1 )e1 e2 .

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CHAPTER 31. BIVECTOR

This can be split into the symmetric, scalar valued, interior product and an antisymmetric, bivector valued exterior product: a · b = a1 b1 + a2 b2 , a ∧ b = (a1 b2 − a2 b1 )e1 e2 = (a1 b2 − a2 b1 )e12 . All bivectors in two dimensions are of this form, that is multiples of the bivector e1 e2 , written e12 to emphasise it is a bivector rather than a vector. The magnitude of e12 is 1, with e212 = −1, so it is called the unit bivector. The term unit bivector can be used in other dimensions but it is only uniquely defined (up to a sign) in two dimensions and all bivectors are multiples of e12 . As the highest grade element of the algebra e12 is also the pseudoscalar which is given the symbol i.

31.4.1

Complex numbers

With the properties of negative square and unit magnitude, the unit bivector can be identified with the imaginary unit from complex numbers. The bivectors and scalars together form the even subalgebra of the geometric algebra, which is isomorphic to the complex numbers ℂ. The even subalgebra has basis (1, e12 ), the whole algebra has basis (1, e1 , e2 , e12 ). The complex numbers are usually identified with the coordinate axes and two-dimensional vectors, which would mean associating them with the vector elements of the geometric algebra. There is no contradiction in this, as to get from a general vector to a complex number an axis needs to be identified as the real axis, e1 say. This multiplies by all vectors to generate the elements of even subalgebra. All the properties of complex numbers can be derived from bivectors, but two are of particular interest. First as with complex numbers products of bivectors and so the even subalgebra are commutative. This is only true in two dimensions, so properties of the bivector in two dimensions that depend on commutativity do not usually generalise to higher dimensions. Second a general bivector can be written

θe12 = iθ, where θ is a real number. Putting this into the Taylor series for the exponential map and using the property e12 2 = −1 results in a bivector version of Euler’s formula,

eθe12 = eiθ = cos θ + i sin θ, which when multiplied by any vector rotates it through an angle θ about the origin: (x′ e1 + y ′ e2 ) = (xe1 + ye2 )eiθ . The product of a vector with a bivector in two dimensions is anticommutative, so the following products all generate the same rotation v′ = veiθ = e−iθ v = e

−iθ 2

iθ

ve 2 .

Of these the last product is the one that generalises into higher dimensions. The quantity needed is called a rotor and is given the symbol R, so in two dimensions a rotor that rotates through angle θ can be written

R=e

−iθ 2

=e

−θe12 2

,

31.5. THREE DIMENSIONS

71

and the rotation it generates is[15]

v′ = RvR−1 .

31.5 Three dimensions In three dimensions the geometric product of two vectors is

ab = (a1 e1 + a2 e2 + a3 e3 )(b1 e1 + b2 e2 + b3 e3 ) = a1 b1 e1 2 + a2 b2 e2 2 + a3 b3 e3 2 + (a2 b3 − a3 b2 )e2 e3 + (a3 b1 − a1 b3 )e3 e1 + (a1 b2 − a2 b1 )e1 e2 . This can be split into the symmetric, scalar valued, interior product and the antisymmetric, bivector valued, exterior product: a · b = a1 b1 + a2 b2 + a3 b3 a ∧ b = (a2 b3 − a3 b2 )e23 + (a3 b1 − a1 b3 )e31 + (a1 b2 − a2 b1 )e12 . In three dimensions all bivectors are simple and so the result of an exterior product. The unit bivectors e23 , e31 and e12 form a basis for the space of bivectors Λ2 ℝ3 , which itself a three-dimensional linear space. So if a general bivector is:

A = A23 e23 + A31 e31 + A12 e12 , they can be added like vectors

A + B = (A23 + B23 )e23 + (A31 + B31 )e31 + (A12 + B12 )e12 . while when multiplied they produce the following

AB = −A23 B23 −A31 B31 −A12 B12 +(A12 B31 −A31 B12 )e23 +(A23 B12 −A12 B23 )e31 +(A31 B23 −A23 B31 )e12 which can be split into symmetric scalar and antisymmetric bivector parts as follows A · B = −A12 B12 − A31 B31 − A23 B23 A × B = (A23 B31 − A31 B23 )e12 + (A12 B23 − A23 B12 )e13 + (A31 B12 − A12 B31 )e23 . The exterior product of two bivectors in three dimensions is zero. A bivector B can be written as the product of its magnitude and a unit bivector, so writing β for |B| and using the Taylor series for the exponential map it can be shown that

B

eB = eβ β = cos β +

B sin β. β

This is another version of Euler’s formula, but with a general bivector in three dimensions. Unlike in two dimensions bivectors are not commutative so properties that depend on commutativity do not apply in three dimensions. For example in general eA + B ≠ eA eB in three (or more) dimensions. The full geometric algebra in three dimensions, Cℓ3(ℝ), has basis (1, e1 , e2 , e3 , e23 , e31 , e12 , e123 ). The element e123 is a trivector and the pseudoscalar for the geometry. Bivectors in three dimensions are sometimes identified with pseudovectors[16] to which they are related, as discussed below.

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CHAPTER 31. BIVECTOR

31.5.1

Quaternions

Bivectors are not closed under the geometric product, but the even subalgebra is. In three dimensions it consists of all scalar and bivector elements of the geometric algebra, so a general element can be written for example a + A, where a is the scalar part and A is the bivector part. It is written Cℓ + 3 and has basis (1, e23 , e31 , e12 ). The product of two general elements of the even subalgebra is

(a + A)(b + B) = ab + aB + bA + A · B + A × B. The even subalgebra, that is the algebra consisting of scalars and bivectors, is isomorphic to the quaternions, ℍ. This can be seen by comparing the basis to the quaternion basis, or from the above product which is identical to the quaternion product, except for a change of sign which relates to the negative products in the bivector interior product A · B. Other quaternion properties can be similarly related to or derived from geometric algebra. This suggests that the usual split of a quaternion into scalar and vector parts would be better represented as a split into scalar and bivector parts; if this is done the quaternion product is merely the geometric product. It also relates quaternions in three dimensions to complex numbers in two, as each is isomorphic to the even subalgebra for the dimension, a relationship that generalises to higher dimensions.

31.5.2

Rotation vector

The rotation vector, from the axis-angle representation of rotations, is a compact way of representing rotations in three dimensions. In its most compact form, it consists of a vector, the product of a unit vector that is the axis of rotation with the (signed) angle of rotation, so that the magnitude of the vector equals the (unsigned) rotation angle. In geometric algebra this vector is represented as a bivector. This can be seen in its relation to quaternions. If the axis is ω and the angle of rotation is θ then the rotation vector is ωθ. The quaternion associated with the rotation is ( ( ) ( )) θ θ q = cos , ω sin 2 2 but this is just the exponential of half of the bivector Ωθ, that is, ( ) ( ) Ωθ e 2 = cos θ2 + Ω sin θ2 So rotation vectors are bivectors, just as quaternions are elements of the geometric algebra, and they are related by the exponential map in that algebra.

31.5.3

Rotors

The bivector Ωθ generates a rotation through the exponential map. The even elements generated rotate a general vector in three dimensions in the same way as quaternions:

v′ = e

Ωθ 2

ve−

Ωθ 2

.

As to two dimensions the quantity eΩθ is called a rotor and written R. The quantity e−Ωθ is then R−1 , and they generate rotations as follows

v′ = RvR−1 . This is identical to two dimensions, except here rotors are four-dimensional objects isomorphic to the quaternions. This can be generalised to all dimensions, with rotors, elements of the even subalgebra with unit magnitude, being generated by the exponential map from bivectors. They form a double cover over the rotation group, so the rotors R and −R represent the same rotation.

31.5. THREE DIMENSIONS

31.5.4

73

Matrices

Bivectors are isomorphic to skew-symmetric matrices; the general bivector B23 e23 + B31 e31 + B12 e12 maps to the matrix 

0  B MB = 12 −B31

−B12 0 B23

 B31 −B23 . 0

This multiplied by vectors on both sides gives the same vector as the product of a vector and bivector; an example is the angular velocity tensor. Skew symmetric matrices generate orthogonal matrices with determinant 1 through the exponential map. In particular the exponent of a bivector associated with a rotation is a rotation matrix, that is the rotation matrix MR given by the above skew-symmetric matrix is

M R = eM B . The rotation described by MR is the same as that described by the rotor R given by B

R=e2, and the matrix MR can be also calculated directly from rotor R: 

(Re1 R−1 ) · e1  MR = (Re1 R−1 ) · e2 (Re1 R−1 ) · e3

(Re2 R−1 ) · e1 (Re2 R−1 ) · e2 (Re2 R−1 ) · e3

 (Re3 R−1 ) · e1 (Re3 R−1 ) · e2 . (Re3 R−1 ) · e3

Bivectors are related to the eigenvalues of a rotation matrix. Given a rotation matrix M the eigenvalues can calculated by solving the characteristic equation for that matrix 0 = det(M − λI). By the fundamental theorem of algebra this has three roots, but only one real root as there is only one eigenvector, the axis of rotation. The other roots must be a complex conjugate pair. They have unit magnitude so purely imaginary logarithms, equal to the magnitude of the bivector associated with the rotation, which is also the angle of rotation. The eigenvectors associated with the complex eigenvalues are in the plane of the bivector, so the exterior product of two non-parallel eigenvectors result in the bivector, or at least a multiple of it.

31.5.5

Axial vectors

The rotation vector is an example of an axial vector. Axial vectors, or pseudovectors, are vectors with the special feature that their coordinates undergo a sign change relative to the usual vectors (also called “polar vectors”) under inversion through the origin, reflection in a plane, or other orientation-reversing linear transformation.[17] Examples include quantities like torque, angular momentum and vector magnetic fields. Quantities that would use axial vectors in vector algebra are properly represented by bivectors in geometric algebra.[18] More precisely, if an underlying orientation is chosen, the axial vectors are naturally identified with the usual vectors; the Hodge dual then gives the isomorphism between axial vectors and bivectors, so each axial vector is associated with a bivector and vice versa; that is

A = ∗a ,

a = ∗A

where ∗ indicates the Hodge dual. Note that if the underlying orientation is reversed by inversion through the origin, both the identification of the axial vectors with the usual vectors and the Hodge dual change sign, but the bivectors don't budge. Alternately, using the unit pseudoscalar in Cℓ3 (ℝ), i = e1 e2 e3 gives

A = ai ,

a = −Ai.

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CHAPTER 31. BIVECTOR

p

x p

x p m

x The 3-angular momentum as a bivector (plane element) and axial vector, of a particle of mass m with instantaneous 3-position x and 3-momentum p.

This is easier to use as the product is just the geometric product. But it is antisymmetric because (as in two dimensions) the unit pseudoscalar i squares to −1, so a negative is needed in one of the products. This relationship extends to operations like the vector valued cross product and bivector valued exterior product, as when written as determinants they are calculated in the same way: e1 a × b = a1 b1

e2 a2 b2

e3 a3 , b3

e23 a ∧ b = a1 b1

e31 a2 b2

e12 a3 , b3

so are related by the Hodge dual:

∗(a ∧ b) = a × b ,

∗(a × b) = a ∧ b.

Bivectors have a number of advantages over axial vectors. They better disambiguate axial and polar vectors, that is the quantities represented by them, so it is clearer which operations are allowed and what their results are. For example the inner product of a polar vector and an axial vector resulting from the cross product in the triple product should result in a pseudoscalar, a result which is more obvious if the calculation is framed as the exterior product of a vector and bivector. They generalises to other dimensions; in particular bivectors can be used to describe quantities like torque and angular momentum in two as well as three dimensions. Also, they closely match geometric intuition in a number of ways, as seen in the next section.[19]

31.5.6

Geometric interpretation

As suggested by their name and that of the algebra, one of the attractions of bivectors is that they have a natural geometric interpretation. This can be described in any dimension but is best done in three where parallels can be drawn with more familiar objects, before being applied to higher dimensions. In two dimensions the geometric

31.5. THREE DIMENSIONS

75

Parallel plane segments with the same orientation and area corresponding to the same bivector a ∧ b.[2]

interpretation is trivial, as the space is two-dimensional so has only one plane, and all bivectors are associated with it differing only by a scale factor. All bivectors can be interpreted as planes, or more precisely as directed plane segments. In three dimensions there are three properties of a bivector that can be interpreted geometrically: • The arrangement of the plane in space, precisely the attitude of the plane (or alternately the rotation, geometric orientation or gradient of the plane), is associated with the ratio of the bivector components. In particular the three basis bivectors, e23 , e31 and e12 , or scalar multiples of them, are associated with the yz-plane, xz-plane and xy-plane respectively.

76

CHAPTER 31. BIVECTOR • The magnitude of the bivector is associated with the area of the plane segment. The area does not have a particular shape so any shape can be used. It can even be represented in other ways, such as by an angular measure. But if the vectors are interpreted as lengths the bivector is usually interpreted as an area with the same units, as follows. • Like the direction of a vector a plane associated with a bivector has a direction, a circulation or a sense of rotation in the plane, which takes two values seen as clockwise and counterclockwise when viewed from viewpoint not in the plane. This is associated with a change of sign in the bivector, that is if the direction is reversed the bivector is negated. Alternately if two bivectors have the same attitude and magnitude but opposite directions then one is the negative of the other. • If imagined as a 2d parallelogram, with vector’s origin at 0, then signed area is the determinant of the vectors’ Cartesian coordinates ( ax by − bx ay ).[20]

a b b a b a The cross product a × b is orthogonal to the bivector a ∧ b.

In three dimensions all bivectors can be generated by the exterior product of two vectors. If the bivector B = a ∧ b then the magnitude of B is

|B| = |a| |b| sin θ, where θ is the angle between the vectors. This is the area of the parallelogram with edges a and b, as shown in the diagram. One interpretation is that the area is swept out by b as it moves along a. The exterior product is

31.5. THREE DIMENSIONS

77

antisymmetric, so reversing the order of a and b to make a move along b results in a bivector with the opposite direction that is the negative of the first. The plane of bivector a ∧ b contains both a and b so they are both parallel to the plane. Bivectors and axial vectors are related by Hodge dual. In a real vector space the Hodge dual relates a subspace to its orthogonal complement, so if a bivector is represented by a plane then the axial vector associated with it is simply the plane’s surface normal. The plane has two normals, one on each side, giving the two possible orientations for the plane and bivector.

Relationship between force F, torque τ, linear momentum p, and angular momentum L.

This relates the cross product to the exterior product. It can also be used to represent physical quantities, like torque and angular momentum. In vector algebra they are usually represented by vectors, perpendicular to the plane of the force, linear momentum or displacement that they are calculated from. But if a bivector is used instead the plane is the plane of the bivector, so is a more natural way to represent the quantities and the way they act. It also unlike the vector representation generalises into other dimensions. The product of two bivectors has a geometric interpretation. For non-zero bivectors A and B the product can be split into symmetric and antisymmetric parts as follows:

AB = A · B + A × B. Like vectors these have magnitudes |A · B| = |A||B| cos θ and |A × B| = |A||B| sin θ, where θ is the angle between the planes. In three dimensions it is the same as the angle between the normal vectors dual to the planes, and it generalises to some extent in higher dimensions. Bivectors can be added together as areas. Given two non-zero bivectors B and C in three dimensions it is always possible to find a vector that is contained in both, a say, so the bivectors can be written as exterior products involving a: B=a∧b C=a∧c

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CHAPTER 31. BIVECTOR

b a

B

c

C

B+C b+c

Two bivectors, two of the non-parallel sides of a prism, being added to give a third bivector.[12]

This can be interpreted geometrically as seen in the diagram: the two areas sum to give a third, with the three areas forming faces of a prism with a, b, c and b + c as edges. This corresponds to the two ways of calculating the area using the distributivity of the exterior product:

B+C=a∧b+a∧c = a ∧ (b + c). This only works in three dimensions as it is the only dimension where a vector parallel to both bivectors must exist. In higher dimensions bivectors generally are not associated with a single plane, or if they are (simple bivectors) two bivectors may have no vector in common, and so sum to a non-simple bivector.

31.6 Four dimensions In four dimensions the basis elements for the space Λ2 ℝ4 of bivectors are (e12 , e13 , e14 , e23 , e24 , e34 ), so a general bivector is of the form

A = a12 e12 + a13 e13 + a14 e14 + a23 e23 + a24 e24 + a34 e34 .

31.6. FOUR DIMENSIONS

31.6.1

79

Orthogonality

In four dimensions bivectors are orthogonal to bivectors. That is, the Hodge dual of a bivector is a bivector, and the space Λ2 ℝ4 is dual to itself in Cℓ4 (ℝ). Normal vectors are not unique, instead every plane is orthogonal to all the vectors in its Hodge dual space. This can be used to partition the bivectors into two 'halves’, for example into two sets of three unit bivectors each. There are only four distinct ways to do this, and whenever it’s done one vector is in only one of the two halves, for example (e12 , e13 , e14 ) and (e23 , e24 , e34 ).

31.6.2

Simple bivectors in 4D

In four dimensions bivectors are generated by the exterior product of vectors in ℝ4 , but with one important difference from ℝ3 and ℝ2 . In four dimensions not all bivectors are simple. There are bivectors such as e12 + e34 that cannot be generated by the external product of two vectors. This also means they do not have a real, that is scalar, square. In this case

(e12 + e34 )2 = e12 e12 + e12 e34 + e34 e12 + e34 e34 = −2 + 2e1234 . The element e1234 is the pseudoscalar in Cℓ4 , distinct from the scalar, so the square is non-scalar. All bivectors in four dimensions can be generated using at most two exterior products and four vectors. The above bivector can be written as

e12 + e34 = e1 ∧ e2 + e3 ∧ e4 . Similarly, every bivector can be written as the sum of two simple bivectors. It is useful to choose two orthogonal bivectors for this, and this is always possible to do. Moreover, for a generic bivector the choice of simple bivectors is unique, that is, there is only one way to decompose into orthogonal bivectors; the only exception is when the two orthogonal bivectors have equal magnitudes (as in the above example): in this case the decomposition is not unique.[1] The decomposition is always unique in the case of simple bivectors, with the added bonus that one of the orthogonal parts is zero.

31.6.3

Rotations in R4

As in three dimensions bivectors in four dimension generate rotations through the exponential map, and all rotations can be generated this way. As in three dimensions if B is a bivector then the rotor R is eB/2 and rotations are generated in the same way:

v ′ = RvR−1 . The rotations generated are more complex though. They can be categorised as follows: simple rotations are those that fix a plane in 4D, and rotate by an angle “about” this plane. double rotations have only one fixed point, the origin, and rotate through two angles about two orthogonal planes. In general the angles are different and the planes are uniquely specified isoclinic rotations are double rotations where the angles of rotation are equal. In this case the planes about which the rotation is taking place are not unique. These are generated by bivectors in a straightforward way. Simple rotations are generated by simple bivectors, with the fixed plane the dual or orthogonal to the plane of the bivector. The rotation can be said to take place about that plane, in the plane of the bivector. All other bivectors generate double rotations, with the two angles of the rotation equalling the magnitudes of the two simple bivectors the non-simple bivector is composed of. Isoclinic rotations arise when these magnitudes are equal, in which case the decomposition into two simple bivectors is not unique.[21]

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CHAPTER 31. BIVECTOR

A 3D projection of an tesseract performing an isoclinic rotation.

Bivectors in general do not commute, but one exception is orthogonal bivectors and exponents of them. So if the bivector B = B1 + B2 , where B1 and B2 are orthogonal simple bivectors, is used to generate a rotation it decomposes into two simple rotations that commute as follows:

R=e

B1 +B2 2

=e

B1 2

e

B2 2

=e

B2 2

e

B1 2

It is always possible to do this as all bivectors can be expressed as sums of orthogonal bivectors.

31.6.4

Spacetime rotations

Spacetime is a mathematical model for our universe used in special relativity. It consists of three space dimensions and one time dimension combined into a single four-dimensional space. It is naturally described using geometric algebra and bivectors, with the Euclidean metric replaced by a Minkowski metric. That is the algebra is identical to that of Euclidean space, except the signature is changed, so { 2

ei =

1, i = 1, 2, 3 −1, i = 4

31.6. FOUR DIMENSIONS

81

(Note the order and indices above are not universal – here e4 is the time-like dimension). The geometric algebra is Cℓ₃,₁(ℝ), and the subspace of bivectors is Λ2 ℝ3,1 . The simple bivectors are of two types. The simple bivectors e23 , e31 and e12 have negative squares and span the bivectors of the three-dimensional subspace corresponding to Euclidean space, ℝ3 . These bivectors generate ordinary rotations in ℝ3 . The simple bivectors e14 , e24 and e34 have positive squares and as planes span a space dimension and the time dimension. These also generate rotations through the exponential map, but instead of trigonometric functions, hyperbolic functions are needed, which generates a rotor as follows:

e

Ωθ 2

( ) ( ) θ θ = cosh + Ω sinh , 2 2

where Ω is the bivector (e14 , etc), identified via the metric with an antisymmetric linear transformation of ℝ3,1 . These are Lorentz boosts, expressed in a particularly compact way, using the same kind of algebra as in ℝ3 and ℝ4 . In general all spacetime rotations are generated from bivectors through the exponential map, that is, a general rotor generated by bivector A is of the form

A

R = e2 . The set of all rotations in spacetime form the Lorentz group, and from them most of the consequences of special relativity can be deduced. More generally this show how transformations in Euclidean space and spacetime can all be described using the same kind of algebra.

31.6.5

Maxwell’s equations

(Note: in this section traditional 3-vectors are indicated by lines over the symbols and spacetime vector and bivectors by bold symbols, with the vectors J and A exceptionally in uppercase) Maxwell’s equations are used in physics to describe the relationship between electric and magnetic fields. Normally given as four differential equations they have a particularly compact form when the fields are expressed as a spacetime bivector from Λ2 ℝ3,1 . If the electric and magnetic fields in ℝ3 are E and B then the electromagnetic bivector is

F=

1 Ee4 + Be123 , c

where e4 is again the basis vector for the time-like dimension and c is the speed of light. The product Be123 yields the bivector that is Hodge dual to B in three dimensions, as discussed above, while Ee4 as a product of orthogonal vectors is also bivector valued. As a whole it is the electromagnetic tensor expressed more compactly as a bivector, and is used as follows. First it is related to the 4-current J, a vector quantity given by

J = j + cρe4 , where j is current density and ρ is charge density. They are related by a differential operator ∂, which is

∂ = ∇ − e4

1 ∂ . c ∂t

The operator ∇ is a differential operator in geometric algebra, acting on the space dimensions and given by ∇M = ∇·M + ∇∧M. When applied to vectors ∇·M is the divergence and ∇∧M is the curl but with a bivector rather than vector result, that is dual in three dimensions to the curl. For general quantity M they act as grade lowering and raising differential operators. In particular if M is a scalar then this operator is just the gradient, and it can be thought of as a geometric algebraic del operator. Together these can be used to give a particularly compact form for Maxwell’s equations in a vacuum:

82

CHAPTER 31. BIVECTOR

∂F = J. This when decomposed according to geometric algebra, using geometric products which have both grade raising and grade lowering effects, is equivalent to Maxwell’s four equations. This is the form in a vacuum, but the general form is only a little more complex. It is also related to the electromagnetic four-potential, a vector A given by

1 A = A + V e4 , c where A is the vector magnetic potential and V is the electric potential. It is related to the electromagnetic bivector as follows

∂A = −F, using the same differential operator ∂.[22]

31.7 Higher dimensions As has been suggested in earlier sections much of geometric algebra generalises well into higher dimensions. The geometric algebra for the real space ℝn is Cℓn(ℝ), and the subspace of bivectors is Λ2 ℝn . The number of simple bivectors needed to form a general bivector rises with the dimension, so for n odd it is (n − 1) / 2, for n even it is n / 2. So for four and five dimensions only two simple bivectors are needed but three are required for six and seven dimensions. For example in six dimensions with standard basis (e1 , e2 , e3 , e4 , e5 , e6 ) the bivector

e12 + e34 + e56 is the sum of three simple bivectors but no less. As in four dimensions it is always possible to find orthogonal simple bivectors for this sum.

31.7.1

Rotations in higher dimensions

As in three and four dimensions rotors are generated by the exponential map, so

B

e2 is the rotor generated by bivector B. Simple rotations, that take place in a plane of rotation around a fixed blade of dimension (n − 2) are generated by simple bivectors, while other bivectors generate more complex rotations which can be described in terms of the simple bivectors they are sums of, each related to a plane of rotation. All bivectors can be expressed as the sum of orthogonal and commutative simple bivectors, so rotations can always be decomposed into a set of commutative rotations about the planes associated with these bivectors. The group of the rotors in n dimensions is the spin group, Spin(n). One notable feature, related to the number of simple bivectors and so rotation planes, is that in odd dimensions every rotation has a fixed axis – it is misleading to call it an axis of rotation as in higher dimensions rotations are taking place in multiple planes orthogonal to it. This is related to bivectors, as bivectors in odd dimensions decompose into the same number of bivectors as the even dimension below, so have the same number of planes, but one extra dimension. As each plane generates rotations in two dimensions in odd dimensions there must be one dimension, that is an axis, that is not being rotated.[23] Bivectors are also related to the rotation matrix in n dimensions. As in three dimensions the characteristic equation of the matrix can be solved to find the eigenvalues. In odd dimensions this has one real root, with eigenvector the fixed axis, and in even dimensions it has no real roots, so either all or all but one of the roots are complex conjugate

31.8. PROJECTIVE GEOMETRY

83

pairs. Each pair is associated with a simple component of the bivector associated with the rotation. In particular the log of each pair is ± the magnitude, while eigenvectors generated from the roots are parallel to and so can be used to generate the bivector. In general the eigenvalues and bivectors are unique, and the set of eigenvalues gives the full decomposition into simple bivectors; if roots are repeated then the decomposition of the bivector into simple bivectors is not unique.

31.8 Projective geometry Geometric algebra can be applied to projective geometry in a straightforward way. The geometric algebra used is Cℓn(ℝ), n ≥ 3, the algebra of the real vector space ℝn . This is used to describe objects in the real projective space ℝℙn − 1 . The non-zero vectors in Cℓn(ℝ) or ℝn are associated with points in the projective space so vectors that differ only by a scale factor, so their exterior product is zero, map to the same point. Non-zero simple bivectors in Λ2 ℝn represent lines in ℝℙn − 1 , with bivectors differing only by a (positive or negative) scale factor representing the same line. A description of the projective geometry can be constructed in the geometric algebra using basic operations. For example given two distinct points in ℝℙn − 1 represented by vectors a and b the line between them is given by a ∧ b (or b ∧ a). Two lines intersect in a point if A ∧ B = 0 for their bivectors A and B. This point is given by the vector p = A ∨ B = (A × B)J −1 . The operation "⋁" is the meet, which can be defined as above in terms of the join, J = A ∧ B for non-zero A ∧ B. Using these operations projective geometry can be formulated in terms of geometric algebra. For example given a third (non-zero) bivector C the point p lies on the line given by C if and only if

p ∧ C = 0. So the condition for the lines given by A, B and C to be collinear is

(A ∨ B) ∧ C = 0, which in Cℓ3 (ℝ) and ℝℙ2 simplifies to ⟨ABC⟩ = 0, where the angle brackets denote the scalar part of the geometric product. In the same way all projective space operations can be written in terms of geometric algebra, with bivectors representing general lines in projective space, so the whole geometry can be developed using geometric algebra.[14]

31.8.1

Tensors and matrices

As noted above a bivector can be written as a skew-symmetric matrix, which through the exponential map generates a rotation matrix that describes the same rotation as the rotor, also generated by the exponential map but applied to the vector. But it is also used with other bivectors such as the angular velocity tensor and the electromagnetic tensor, respectively a 3×3 and 4×4 skew-symmetric matrix or tensor. Real bivectors in Λ2 ℝn are isomorphic to n×n skew-symmetric matrices, or alternately to antisymmetric tensors of order 2 on ℝn . While bivectors are isomorphic to vectors (via the dual) in three dimensions they can be represented by skew-symmetric matrices in any dimension. This is useful for relating bivectors to problems described by matrices, so they can be re-cast in terms of bivectors, given a geometric interpretation, then often solved more easily or related geometrically to other bivector problems.[24] More generally every real geometric algebra is isomorphic to a matrix algebra. These contain bivectors as a subspace, though often in a way which is not especially useful. These matrices are mainly of interest as a way of classifying Clifford algebras.[25]

84

CHAPTER 31. BIVECTOR

31.9 See also • p-vector • Multilinear algebra

31.10 Notes [1] Lounesto (2001) p. 87 [2] Leo Dorst, Daniel Fontijne, Stephen Mann (2009). Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry (2nd ed.). Morgan Kaufmann. p. 32. ISBN 0-12-374942-5. The algebraic bivector is not specific on shape; geometrically it is an amount of oriented area in a specific plane, that’s all. [3] David Hestenes (1999). New foundations for classical mechanics: Fundamental Theories of Physics (2nd ed.). Springer. p. 21. ISBN 0-7923-5302-1. [4] Lounesto (2001) p. 33 [5] Karen Hunger Parshall, David E. Rowe (1997). The Emergence of the American Mathematical Research Community, 1876– 1900. American Mathematical Society. p. 31 ff. ISBN 0-8218-0907-5. [6] Rida T. Farouki (2007). “Chapter 5: Quaternions”. Pythagorean-hodograph curves: algebra and geometry inseparable. Springer. p. 60 ff. ISBN 3-540-73397-3. [7] A discussion of quaternions from these years is Alexander McAulay (1911). “Quaternions”. The encyclopædia britannica: a dictionary of arts, sciences, literature and general information. Vol. 22 (11th ed.). Cambridge University Press. p. 718 et seq. [8] Josiah Willard Gibbs, Edwin Bidwell Wilson (1901). Vector analysis: a text-book for the use of students of mathematics and physics. Yale University Press. p. 481 ff. [9] Philippe Boulanger, Michael A. Hayes (1993). Bivectors and waves in mechanics and optics. Springer. ISBN 0-412-464608. [10] PH Boulanger & M Hayes (1991). “Bivectors and inhomogeneous plane waves in anisotropic elastic bodies”. In Julian J. Wu, Thomas Chi-tsai Ting, David M. Barnett. Modern theory of anisotropic elasticity and applications. Society for Industrial and Applied Mathematics (SIAM). p. 280 et seq. ISBN 0-89871-289-0. [11] David Hestenes. op. cit. p. 61. ISBN 0-7923-5302-1. [12] Lounesto (2001) p. 35 [13] Lounesto (2001) p. 86 [14] Hestenes, David; Ziegler, Renatus (1991). “Projective Geometry with Clifford Algebra” (PDF). Acta Applicandae Mathematicae 23: 25–63. doi:10.1007/bf00046919. [15] Lounesto (2001) p.29 [16] William E Baylis (1994). Theoretical methods in the physical sciences: an introduction to problem solving using Maple V. Birkhäuser. p. 234, see footnote. ISBN 0-8176-3715-X. The terms axial vector and pseudovector are often treated as synonymous, but it is quite useful to be able to distinguish a bivector (...the pseudovector) from its dual (...the axial vector). [17] In strict mathematical terms, axial vectors are an n-dimensional vector space equipped with the usual structure group GL(n,R), but with the nonstandard representation A → A det(A)/|det(A)|. [18] Chris Doran, Anthony Lasenby (2003). Geometric algebra for physicists. Cambridge University Press. p. 56. ISBN 0-521-48022-1. [19] Lounesto (2001) pp. 37–39 [20] WildLinAlg episode 4, Norman J Wildberger, Univ. of New South Wales, 2010, lecture via youtube [21] Lounesto (2001) pp. 89–90 [22] Lounesto (2001) pp. 109–110 [23] Lounesto (2001) p.222 [24] Lounesto (2001) p. 193 [25] Lounesto (2001) p. 217

31.11. GENERAL REFERENCES

85

31.11 General references • Leo Dorst, Daniel Fontijne, Stephen Mann (2009). "§ 2.3.3 Visualizing bivectors”. Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry (2nd ed.). Morgan Kaufmann. p. 31 ff. ISBN 0-12-374942-5. • Whitney, Hassler (1957). Geometric Integration Theory. Princeton: Princeton University Press. ISBN 978-0486-44583-0. • Lounesto, Pertti (2001). Clifford algebras and spinors. Cambridge: Cambridge University Press. ISBN 9780-521-00551-7. • Chris Doran and Anthony Lasenby (2003). "§ 1.6 The outer product”. Geometric Algebra for Physicists. Cambridge: Cambridge University Press. p. 11 et seq. ISBN 978-0-521-71595-9.

Chapter 32

Blade (geometry) In geometric algebra, a blade is a generalization of the concept of scalars and vectors to include simple bivectors, trivectors, etc. Specifically, a k-blade is any object that can be expressed as the exterior product (informally wedge product) of k vectors, and is of grade k. In detail:[1] • A 0-blade is a scalar. • A 1-blade is a vector. Every vector is simple. • A 2-blade is a simple bivector. Linear combinations of 2-blades also are bivectors, but need not be simple, and are hence not necessarily 2-blades. A 2-blade may be expressed as the wedge product of two vectors a and b: a ∧ b. • A 3-blade is a simple trivector, that is, it may expressed as the wedge product of three vectors a, b, and c: a ∧ b ∧ c. • In a space of dimension n, a blade of grade n − 1 is called a pseudovector.[2] • The highest grade element in a space is called a pseudoscalar, and in a space of dimension n is an n-blade.[3] • In a space of dimension n, there are k(n − k) + 1 dimensions of freedom in choosing a k-blade, of which one dimension is an overall scaling multiplier.[4] For an n-dimensional space, there are blades of all grades from 0 to n inclusive. A vector subspace of finite dimension k may be represented by the k-blade formed as a wedge product of all the elements of a basis for that subspace.[5]

32.1 Examples For example, in 2-dimensional space scalars are described as 0-blades, vectors are 1-blades, and area elements are 2-blades known as pseudoscalars, in that they are one-dimensional objects distinct from regular scalars. In three-dimensional space, 0-blades are again scalars and 1-blades are three-dimensional vectors, but in threedimensions, areas have an orientation, so while 2-blades are area elements, they are oriented. 3-blades (trivectors) represent volume elements and in three-dimensional space, these are scalar-like—i.e., 3-blades in three-dimensions form a one-dimensional vector space.

32.2 See also • Multivector 86

32.3. NOTES

87

• Exterior algebra • Geometric algebra • Clifford algebra

32.3 Notes [1] Marcos A. Rodrigues (2000). "§1.2 Geometric algebra: an outline”. Invariants for pattern recognition and classification. World Scientific. p. 3 ff. ISBN 981-02-4278-6. [2] William E Baylis (2004). "§4.2.3 Higher-grade multivectors in Cℓn: Duals”. Lectures on Clifford (geometric) algebras and applications. Birkhäuser. p. 100. ISBN 0-8176-3257-3. [3] John A. Vince (2008). Geometric algebra for computer graphics. Springer. p. 85. ISBN 1-84628-996-3. [4] For Grassmannians (including the result about dimension) a good book is: Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-05059-9, MR 1288523. The proof of the dimensionality is actually straightforward. Take k vectors and wedge them together v1 ∧· · ·∧vk and perform elementary column operations on these (factoring the pivots out) until the top k × k block are elementary basis vectors of Fk . The wedge product is then parametrized by the product of the pivots and the lower k × (n − k) block. [5] David Hestenes (1999). New foundations for classical mechanics: Fundamental Theories of Physics. Springer. p. 54. ISBN 0-7923-5302-1.

32.4 General references • David Hestenes, Garret Sobczyk (1987). “Chapter 1: Geometric algebra”. Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. Springer. p. 1 ff. ISBN 90-277-2561-6. • Chris Doran and Anthony Lasenby (2003). Geometric algebra for physicists. Cambridge University Press. ISBN 0-521-48022-1. • A Lasenby, J Lasenby & R Wareham (2004) A covariant approach to geometry using geometric algebra Technical Report. University of Cambridge Department of Engineering, Cambridge, UK. • R Wareham, J Cameron, & J Lasenby (2005). “Applications of conformal geometric algebra to computer vision and graphics”. In Hongbo Li, Peter J. Olver, Gerald Sommer. Computer algebra and geometric algebra with applications. Springer. p. 329 ff. ISBN 3-540-26296-2.

32.5 External links • A Geometric Algebra Primer, especially for computer scientists.

Chapter 33

Boolean ring In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R,[1][2][3] such as the ring of integers modulo 2. That is, R consists only of idempotent elements.[4][5] Every Boolean ring gives rise to a Boolean algebra, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨).

33.1 Notations There are at least four different and incompatible systems of notation for Boolean rings and algebras. • In commutative algebra the standard notation is to use x + y = (x ∧ ¬ y) ∨ (¬ x ∧ y) for the ring sum of x and y, and use xy = x ∧ y for their product. • In logic, a common notation is to use x ∧ y for the meet (same as the ring product) and use x ∨ y for the join, given in terms of ring notation (given just above) by x + y + xy. • In set theory and logic it is also common to use x · y for the meet, and x + y for the join x ∨ y. This use of + is different from the use in ring theory. • A rare convention is to use xy for the product and x ⊕ y for the ring sum, in an effort to avoid the ambiguity of +. The old terminology was to use “Boolean ring” to mean a “Boolean ring possibly without an identity”, and “Boolean algebra” to mean a Boolean ring with an identity. (This is the same as the old use of the terms “ring” and “algebra” in measure theory) (Also note that, when a Boolean ring has an identity, then a complement operation becomes definable on it, and a key characteristic of the modern definitions of both Boolean algebra and sigma-algebra is that they have complement operations.)

33.2 Examples One example of a Boolean ring is the power set of any set X, where the addition in the ring is symmetric difference, and the multiplication is intersection. As another example, we can also consider the set of all finite or cofinite subsets of X, again with symmetric difference and intersection as operations. More generally with these operations any field of sets is a Boolean ring. By Stone’s representation theorem every Boolean ring is isomorphic to a field of sets (treated as a ring with these operations).

33.3 Relation to Boolean algebras Since the join operation ∨ in a Boolean algebra is often written additively, it makes sense in this context to denote ring addition by ⊕, a symbol that is often used to denote exclusive or. 88

33.4. PROPERTIES OF BOOLEAN RINGS

x

y

89

x

x⋀y

y x⋁y

x ¬x

Venn diagrams for the Boolean operations of conjunction, disjunction, and complement

Given a Boolean ring R, for x and y in R we can define x ∧ y = xy, x ∨ y = x ⊕ y ⊕ xy, ¬x = 1 ⊕ x. These operations then satisfy all of the axioms for meets, joins, and complements in a Boolean algebra. Thus every Boolean ring becomes a Boolean algebra. Similarly, every Boolean algebra becomes a Boolean ring thus: xy = x ∧ y, x ⊕ y = (x ∨ y) ∧ ¬(x ∧ y). If a Boolean ring is translated into a Boolean algebra in this way, and then the Boolean algebra is translated into a ring, the result is the original ring. The analogous result holds beginning with a Boolean algebra. A map between two Boolean rings is a ring homomorphism if and only if it is a homomorphism of the corresponding Boolean algebras. Furthermore, a subset of a Boolean ring is a ring ideal (prime ring ideal, maximal ring ideal) if and only if it is an order ideal (prime order ideal, maximal order ideal) of the Boolean algebra. The quotient ring of a Boolean ring modulo a ring ideal corresponds to the factor algebra of the corresponding Boolean algebra modulo the corresponding order ideal.

33.4 Properties of Boolean rings Every Boolean ring R satisfies x ⊕ x = 0 for all x in R, because we know x ⊕ x = (x ⊕ x)2 = x2 ⊕ x2 ⊕ x2 ⊕ x2 = x ⊕ x ⊕ x ⊕ x and since is an abelian group, we can subtract x ⊕ x from both sides of this equation, which gives x ⊕ x = 0. A similar proof shows that every Boolean ring is commutative: x ⊕ y = (x ⊕ y)2 = x2 ⊕ xy ⊕ yx ⊕ y2 = x ⊕ xy ⊕ yx ⊕ y and this yields xy ⊕ yx = 0, which means xy = yx (using the first property above). The property x ⊕ x = 0 shows that any Boolean ring is an associative algebra over the field F2 with two elements, in just one way. In particular, any finite Boolean ring has as cardinality a power of two. Not every associative algebra with one over F2 is a Boolean ring: consider for instance the polynomial ring F2 [X].

90

CHAPTER 33. BOOLEAN RING

The quotient ring R/I of any Boolean ring R modulo any ideal I is again a Boolean ring. Likewise, any subring of a Boolean ring is a Boolean ring. Every prime ideal P in a Boolean ring R is maximal: the quotient ring R/P is an integral domain and also a Boolean ring, so it is isomorphic to the field F2 , which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings. Boolean rings are von Neumann regular rings. Boolean rings are absolutely flat: this means that every module over them is flat. Every finitely generated ideal of a Boolean ring is principal (indeed, (x,y)=(x+y+xy)). Unification in Boolean rings is decidable,[6][7] that is, algorithms exist to solve arbitrary equations over Boolean rings.

33.5 Notes [1] Fraleigh (1976, p. 200) [2] Herstein (1964, p. 91) [3] McCoy (1968, p. 46) [4] Fraleigh (1976, p. 25) [5] Herstein (1964, p. 224) [6] Martin, U., Nipkow, T. (1986). “Unification in Boolean Rings”. In Jörg H. Siekmann. Proc. 8th CADE. LNCS 230. Springer. pp. 506–513. [7] A. Boudet, J.-P. Jouannaud, M. Schmidt-Schauß (1989). “Unification of Boolean Rings and Abelian Groups” (PDF). Journal of Symbolic Computation 8: 449–477. doi:10.1016/s0747-7171(89)80054-9.

33.6 References • Atiyah, Michael Francis; Macdonald, I. G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8 • Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: Addison-Wesley, ISBN 0201-01984-1 • Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN 978-1114541016 • McCoy, Neal H. (1968), Introduction To Modern Algebra (Revised ed.), Boston: Allyn and Bacon, LCCN 68015225 • Ryabukhin, Yu. M. (2001), “Boolean_ring”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

33.7 External links • John Armstrong, Boolean Rings

Chapter 34

Brauer group In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is named after the algebraist Richard Brauer. The group may also be defined in terms of Galois cohomology. More generally, the Brauer group of a scheme is defined in terms of Azumaya algebras.

34.1 Construction A central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A, which is a simple ring, and for which the center is exactly K. Note that CSAs are in general not division algebras, though CSAs can be used to classify division algebras. For example, the complex numbers C form a CSA over themselves, but not over R (the center is C itself, hence too large to be CSA over R). The finite-dimensional division algebras with center R (that means the dimension over R is finite) are the real numbers and the quaternions by a theorem of Frobenius, while any matrix ring over the reals or quaternions – M(n,R) or M(n,H) – is a CSA over the reals, but not a division algebra (if n > 1 ). We obtain an equivalence relation on CSAs over K by the Artin–Wedderburn theorem (Wedderburn's part, in fact), to express any CSA as a M(n,D) for some division algebra D. If we look just at D, that is, if we impose an equivalence relation identifying M(m,D) with M(n,D) for all integers m and n at least 1, we get the Brauer equivalence and the Brauer classes. Given central simple algebras A and B, one can look at the their tensor product A ⊗ B as a K-algebra (see tensor product of R-algebras). It turns out that this is always central simple. A slick way to see this is to use a characterisation: a central simple algebra over K is a K-algebra that becomes a matrix ring when we extend the field of scalars to an algebraic closure of K. Given this closure property for CSAs, they form a monoid under tensor product, compatible with Brauer equivalence, and the Brauer classes are all invertible: the inverse class to that of an algebra A is the one containing the opposite algebra Aop (the opposite ring with the same action by K since the image of K → A is in the center of A). In other words, for a CSA A we have A ⊗ Aop = M(n2 ,K), where n is the degree of A over K. (This provides a substantial reason for caring about the notion of an opposite algebra: it provides the inverse in the Brauer group.)

34.2 Examples • In the following cases, every finite-dimensional central division algebra over a field K is K itself, so that the Brauer group Br(K) is trivial: •

• K is an algebraically closed field:[1] more generally, this is true for any pseudo algebraically closed field[2] or quasi-algebraically closed field.[3]

•

• K is a finite field (Wedderburn’s theorem);[1][4] 91

92

CHAPTER 34. BRAUER GROUP •

• K is the function field of an algebraic curve over an algebraically closed field (Tsen’s theorem);[4]

•

• An algebraic extension of Q containing all roots of unity.[4]

• The Brauer group Br(R) of the field R of real numbers is the cyclic group of order two. There are just two non-isomorphic real division algebras with center R: the algebra R itself and the quaternion algebra H.[5] Since H ⊗ H ≅ M(4,R), the class of H has order two in the Brauer group. More generally, any real closed field has Brauer group of order two.[1] • K is complete under a discrete valuation with finite residue field. Br(K) is isomorphic to Q/Z.[5]

34.3 Brauer group and class field theory The notion of Brauer group plays an important role in the modern formulation of the class field theory. If Kv is a non-archimedean local field, the Hasse invariants gives a canonical isomorphism invv: Br(Kv) → Q/Z constructed in local class field theory.[6][7][8] An element of the Brauer group of order n can be represented by a cyclic division algebra of dimension n2 .[9] The case of a global field K is addressed by the global class field theory. If D is a central simple algebra over K and v is a valuation then D ⊗ Kv is a central simple algebra over Kv, the local completion of K at v. This defines a homomorphism from the Brauer group of K into the Brauer group of Kv. A given central simple algebra D splits for all but finitely many v, so that the image of D under almost all such homomorphisms is 0. The Brauer group Br(K) fits into an exact sequence[5][10] 0 → Br(K) →

⊕

Br(Kv ) → Q/Z → 0,

v∈S

where S is the set of all valuations of K and the right arrow is the direct sum of the local invariants: the Brauer group of the real numbers is identified with (1/2)Z/Z. The injectivity of the left arrow is the content of the Albert–Brauer– Hasse–Noether theorem. Exactness in the middle term is a deep fact from the global class field theory. The group Q/Z on the right may be interpreted as the “Brauer group” of the class formation of idele classes associated to K.

34.4 Properties • Base change from a field K to an extension field L gives a restriction map from Br(K) to Br(L). The kernel is the group Br(L/K) of classes of K-algebras that split over L. • The Brauer group of any field is a torsion group.[11]

34.5 General theory For an arbitrary field K, the Brauer group may be expressed in terms of Galois cohomology as follows:[12] ∗ Br(K) ∼ = H 2 (Gal(K s /K), K s ).

Here, K s is the separable closure of K, which coincides with the algebraic closure when K is a perfect field. Note that every finite dimensional central simple algebra has a separable splitting field.[13] The isomorphism of the Brauer group with a Galois cohomology group can be described as follows. If D is a division algebra over K of dimension n2 containing a Galois extension L of degree n over K, then the subgroup of elements of D* that normalize L is an extension of the Galois group Gal(L/K) by the nonzero elements L* of L, so corresponds to an element of H2 (Gal(L/K), L*). A generalisation of the Brauer group to the case of commutative rings was introduced by Maurice Auslander and Oscar Goldman,[14] and more generally for schemes by Alexander Grothendieck. In their approach, central simple algebras over a field are replaced with Azumaya algebras.[15]

34.6. SEE ALSO

93

34.6 See also • Algebraic K-theory

34.7 Notes [1] Lorenz (2008) p.164 [2] Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 11 (3rd revised ed.). Springer-Verlag. p. 209. ISBN 978-3-540-77269-9. Zbl 1145.12001. [3] Serre (1979) p.161 [4] Serre (1979) p.162 [5] Serre (1979) p.163 [6] Lorenz (2008) p.232 [7] Serre (1967) p.137 [8] Shatz (1972) p.155 [9] Lorenz (2008) p.226 [10] Gille & Szamuely (2006) p.159 [11] Lorenz (2008) p.194 [12] Serre (1979) pp.157-159 [13] Jacobson (1996) p.93 [14] Auslander, Maurice; Goldman, Oscar (1961). “The Brauer group of a commutative ring”. Trans. Am. Math. Soc. 97: 367–409. doi:10.1090/s0002-9947-1960-0121392-6. ISSN 0002-9947. Zbl 0100.26304. [15] Saltman (1999) p.21

34.8 References • V.A. Iskovskikh (2001), “Brauer group of a field k", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. ISBN 978-0-387-72487-4. Zbl 1130.12001. • Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001. • Jacobson, Nathan (1996). Finite-dimensional division algebras over fields. Berlin: Springer-Verlag. ISBN 3-540-57029-2. Zbl 0874.16002. • Pierce, Richard (1982). Associative algebras. Graduate Texts in Mathematics 88. New York-Berlin: SpringerVerlag. ISBN 0-387-90693-2. Zbl 0497.16001. • Reiner, I. (2003). Maximal Orders. London Mathematical Society Monographs. New Series 28. Oxford University Press. pp. 237–241. ISBN 0-19-852673-3. Zbl 1024.16008. • Saltman, David J. (1999). Lectures on division algebras. Regional Conference Series in Mathematics 94. Providence, RI: American Mathematical Society. ISBN 0-8218-0979-2. Zbl 0934.16013.

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CHAPTER 34. BRAUER GROUP • Serre, Jean-Pierre (1967). “VI. Local class field theory”. In Cassels, J.W.S.; Fröhlich, A. Algebraic number theory. Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union. London: Academic Press. pp. 128–161. Zbl 0153.07403. • Serre, Jean-Pierre (1979). Local Fields. Graduate Texts in Mathematics 67. Translated from the French by Marvin Jay Greenberg. Springer-Verlag. ISBN 0-387-90424-7. Zbl 0423.12016. • Shatz, Stephen S. (1972). Profinite groups, arithmetic, and geometry. Annals of Mathematics Studies 67. Princeton, NJ: Princeton University Press. ISBN 0-691-08017-8. MR 0347778. Zbl 0236.12002.

34.9 Further reading • DeMeyer, F.; Ingraham, E. (1971). Separable algebras over commutative rings. Lecture Notes in Mathematics 181. Berlin-Heidelberg-New York: Springer-Verlag. ISBN 978-3-540-05371-2. Zbl 0215.36602.

34.10 External links • PlanetMath page • MathWorld page

Chapter 35

Buchsbaum ring In mathematics, Buchsbaum rings are Noetherian local rings such that every system of parameters is a weak sequence. They were introduced by Jürgen Stückrad and Wolfgang Vogel (1973) and are named after David Buchsbaum. Every Cohen–Macaulay local ring is a Buchsbaum ring.

35.1 References • Buchsbaum, D. (1966), “Complexes in local ring theory”, in Herstein, I. N., Some aspects of ring theory, Centro Internazionale Matematico Estivo (C.I.M.E.). II Ciclo, Varenna (Como), 23-31 agosto 1965, Rome: Edizioni cremonese, pp. 223–228, ISBN 978-3-642-11035-1, MR 0223393 • Goto (, Shiro (2001), “Buchsbaum_ring”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Stückrad, Jürgen; Vogel, Wolfgang (1973), “Eine Verallgemeinerung der Cohen-Macaulay Ringe und Anwendungen auf ein Problem der Multiplizitätstheorie”, Journal of Mathematics of Kyoto University 13: 513–528, ISSN 0023-608X, MR 0335504 • Stückrad, Jürgen; Vogel, Wolfgang (1986), Buchsbaum rings and applications, Berlin, New York: SpringerVerlag, ISBN 978-3-540-16844-7, MR 881220

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Bézout domain In mathematics, a Bézout domain is a form of a Prüfer domain. It is an integral domain in which the sum of two principal ideals is again a principal ideal. This means that for every pair of elements a Bézout identity holds, and that every finitely generated ideal is principal. Any principal ideal domain (PID) is a Bézout domain, but a Bézout domain need not be a Noetherian ring, so it could have non-finitely generated ideals (which obviously excludes being a PID); if so, it is not a unique factorization domain (UFD), but still is a GCD domain. The theory of Bézout domains retains many of the properties of PIDs, without requiring the Noetherian property. Bézout domains are named after the French mathematician Étienne Bézout.

36.1 Examples • All PIDs are Bézout domains. • Examples of Bézout domains that are not PIDs include the ring of entire functions (functions holomorphic on the whole complex plane) and the ring of all algebraic integers.[1] In case of entire functions, the only irreducible elements are functions associated to a polynomial function of degree 1, so an element has a factorization only if it has finitely many zeroes. In the case of the algebraic integers there are no irreducible elements at all, since for any algebraic integer its square root (for instance) is also an algebraic integer. This shows in both cases that the ring is not a UFD, and so certainly not a PID. • Valuation rings are Bézout domains. Any non-Noetherian valuation ring is an example of a non-noetherian Bézout domain. • The following general construction produces a Bézout domain S that is not a UFD from any Bézout domain R that is not a field, for instance from a PID; the case R = Z is the basic example to have in mind. Let F be the field of fractions of R, and put S = R + XF[X], the subring of polynomials in F[X] with constant term in R. This ring is not Noetherian, since an element like X with zero constant term can be divided indefinitely by noninvertible elements of R, which are still noninvertible in S, and the ideal generated by all these quotients of is not finitely generated (and so X has no factorization in S). One shows as follows that S is a Bézout domain. 1. It suffices to prove that for every pair a, b in S there exist s,t in S such that as + bt divides both a and b. 2. If a and b have a common divisor d, it suffices to prove this for a/d and b/d, since the same s,t will do. 3. We may assume the polynomials a and b nonzero; if both have a zero constant term, then let n be the minimal exponent such that at least one of them has a nonzero coefficient of Xn ; one can find f in F such that fXn is a common divisor of a and b and divide by it. 4. We may therefore assume at least one of a, b has a nonzero constant term. If a and b viewed as elements of F[X] are not relatively prime, there is a greatest common divisor of a and b in this UFD that has constant term 1, and therefore lies in S; we can divide by this factor. 96

36.2. PROPERTIES

97

5. We may therefore also assume that a and b are relatively prime in F[X], so that 1 lies in aF[X] + bF[X], and some constant polynomial r in R lies in aS + bS. Also, since R is a Bézout domain, the gcd d in R of the constant terms a0 and b0 lies in a0 R + b0 R. Since any element without constant term, like a −a0 or b − b0 , is divisible by any nonzero constant, the constant d is a common divisor in S of a and b; we shall show it is in fact a greatest common divisor by showing that it lies in aS + bS. Multiplying a and b respectively by the Bézout coefficients for d with respect to a0 and b0 gives a polynomial p in aS + bS with constant term d. Then p − d has a zero constant term, and so is a multiple in S of the constant polynomial r, and therefore lies in aS + bS. But then d does as well, which completes the proof.

36.2 Properties A ring is a Bézout domain if and only if it is an integral domain in which any two elements have a greatest common divisor that is a linear combination of them: this is equivalent to the statement that an ideal which is generated by two elements is also generated by a single element, and induction demonstrates that all finitely generated ideals are principal. The expression of the greatest common divisor of two elements of a PID as a linear combination is often called Bézout’s identity, whence the terminology. Note that the above gcd condition is stronger than the mere existence of a gcd. An integral domain where a gcd exists for any two elements is called a GCD domain and thus Bézout domains are GCD domains. In particular, in a Bézout domain, irreducibles are prime (but as the algebraic integer example shows, they need not exist). For a Bézout domain R, the following conditions are all equivalent: 1. R is a principal ideal domain. 2. R is Noetherian. 3. R is a unique factorization domain (UFD). 4. R satisfies the ascending chain condition on principal ideals (ACCP). 5. Every nonzero nonunit in R factors into a product of irreducibles (R is an atomic domain). The equivalence of (1) and (2) was noted above. Since a Bézout domain is a GCD domain, it follows immediately that (3), (4) and (5) are equivalent. Finally, if R is not Noetherian, then there exists an infinite ascending chain of finitely generated ideals, so in a Bézout domain an infinite ascending chain of principal ideals. (4) and (2) are thus equivalent. A Bézout domain is a Prüfer domain, i.e., a domain in which each finitely generated ideal is invertible, or said another way, a commutative semihereditary domain.) Roughly speaking, one may view the implications “Bézout domain implies Prüfer domain and GCD-domain” as the non-Noetherian analogues of the more familiar “PID implies Dedekind domain and UFD”. The analogy fails to be precise in that a UFD (or an atomic Prüfer domain) need not be Noetherian. Prüfer domains can be characterized as integral domains whose localizations at all prime (equivalently, at all maximal) ideals are valuation domains. So the localization of a Bézout domain at a prime ideal is a valuation domain. Since an invertible ideal in a local ring is principal, a local ring is a Bézout domain iff it is a valuation domain. Moreover a valuation domain with noncyclic (equivalently non-discrete) value group is not Noetherian, and every totally ordered abelian group is the value group of some valuation domain. This gives many examples of non-Noetherian Bézout domains. In noncommutative algebra, right Bézout domains are domains whose finitely generated right ideals are principal right ideals, that is, of the form xR for some x in R. One notable result is that a right Bézout domain is a right Ore domain. This fact is not interesting in the commutative case, since every commutative domain is an Ore domain. Right Bézout domains are also right semihereditary rings.

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36.3 Modules over a Bézout domain Some facts about modules over a PID extend to modules over a Bézout domain. Let R be a Bézout domain and M finitely generated module over R. Then M is flat if and only if it is torsion-free.[2]

36.4 See also • Semifir (a commutative semifir is precisely a Bézout domain.) • Bézout ring

36.5 References [1] Cohn [2] Bourbaki 1989, Ch I, §2, no 4, Proposition 3

• Cohn, P. M. (1968), “Bezout rings and their subrings”, Proc. Cambridge Philos. Soc. 64: 251–264, doi:10.1017/s03050041000427 MR 0222065 (36 #5117) • Helmer, Olaf (1940), “Divisibility properties of integral functions”, Duke Math. J. 6: 345–356, doi:10.1215/s00127094-40-00626-3, ISSN 0012-7094, MR 0001851 (1,307c) • Kaplansky, Irving (1970), Commutative rings, Boston, Mass.: Allyn and Bacon Inc., pp. x+180, MR 0254021 (40 #7234) • Bourbaki, Nicolas (1989), Commutative algebra • Hazewinkel, Michiel, ed. (2001), “Bezout ring”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608010-4

Chapter 37

C-semiring A c-semiring is a semiring with idempotent addition.

99

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Cartan–Brauer–Hua theorem In abstract algebra, the Cartan–Brauer–Hua theorem (named after Richard Brauer, Élie Cartan, and Hua Luogeng) is a theorem pertaining to division rings. It says that given two division rings K ⊆ D such that xKx−1 is contained in K for every x not equal to 0 in D, either K is contained in the center of D, or K = D. In other words, if the unit group of K is a normal subgroup of the unit group of D, then either K = D or K is central (Lam 2001, p. 211).

38.1 References • Herstein, I. N. (1975). Topics in algebra. New York: Wiley. p. 368. ISBN 0-471-01090-1. • Lam, Tsit-Yuen (2001). A First Course in Noncommutative Rings (2nd ed.). Berlin, New York: SpringerVerlag. ISBN 978-0-387-95325-0. MR 1838439.

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Category of rings In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (preserving the identity). Like many categories in mathematics, the category of rings is large, meaning that the class of all rings is proper.

39.1 As a concrete category The category Ring is a concrete category meaning that the objects are sets with additional structure (addition and multiplication) and the morphisms are functions preserving this structure. There is a natural forgetful functor U : Ring → Set for the category of rings to the category of sets which sends each ring to its underlying set (thus “forgetting” the operations of addition and multiplication). This functor has a left adjoint F : Set → Ring which assigns to each set X the free ring generated by X. One can also view the category of rings as a concrete category over Ab (the category of abelian groups) or over Mon (the category of monoids). Specifically, there are faithful functors A : Ring → Ab M : Ring → Mon which “forget” multiplication and addition, respectively. Both of these functors have left adjoints. The left adjoint of A is the functor which assigns to every abelian group X (thought of as a Z-module) the tensor ring T(X). The left adjoint of M is the functor which assigns to every monoid X the integral monoid ring Z[M].

39.2 Properties 39.2.1

Limits and colimits

The category Ring is both complete and cocomplete, meaning that all small limits and colimits exist in Ring. Like many other algebraic categories, the forgetful functor U : Ring → Set creates (and preserves) limits and filtered colimits, but does not preserve either coproducts or coequalizers. The forgetful functors to Ab and Mon also create and preserve limits. Examples of limits and colimits in Ring include: 101

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• The ring of integers Z is an initial object in Ring. • The zero ring is a terminal object in Ring. • The product in Ring is given by the direct product of rings. This is just the cartesian product of the underlying sets with addition and multiplication defined component-wise. • The coproduct of a family of rings exists and is given by a construction analogous to the free product of groups. The coproduct of nonzero rings can be the zero ring; in particular, this happens whenever the factors have relatively prime characteristic (since the characteristic of the coproduct of (Ri)i∈I must divide the characteristics of each of the rings Ri). • The equalizer in Ring is just the set-theoretic equalizer (the equalizer of two ring homomorphisms is always a subring). • The coequalizer of two ring homomorphisms f and g from R to S is the quotient of S by the ideal generated by all elements of the form f(r) − g(r) for r ∈ R. • Given a ring homomorphism f : R → S the kernel pair of f (this is just the pullback of f with itself) is a congruence relation on R. The ideal determined by this congruence relation is precisely the (ring-theoretic) kernel of f. Note that category-theoretic kernels do not make sense in Ring since there are no zero morphisms (see below). • The ring of p-adic integers is the inverse limit in Ring of a sequence of rings of integers mod pn .

39.2.2

Morphisms

Unlike many categories studied in mathematics, there do not always exist morphisms between pairs of objects in Ring. This is a consequence of the fact that ring homomorphisms must preserve the identity. For example, there are no morphisms from the zero ring 0 to any nonzero ring. A necessary condition for there to be morphisms from R to S is that the characteristic of S divide that of R. Note that even though some of the hom-sets are empty, the category Ring is still connected since it has an initial object. Some special classes of morphisms in Ring include: • Isomorphisms in Ring are the bijective ring homomorphisms. • Monomorphisms in Ring are the injective homomorphisms. Not every monomorphism is regular however. • Every surjective homomorphism is an epimorphism in Ring, but the converse is not true. The inclusion Z → Q is a nonsurjective epimorphism. The natural ring homomorphism from any commutative ring R to any one of its localizations is an epimorphism which is not necessarily surjective. • The surjective homomorphisms can be characterized as the regular or extremal epimorphisms in Ring (these two classes coinciding). • Bimorphisms in Ring are the injective epimorphisms. The inclusion Z → Q is an example of a bimorphism which is not an isomorphism.

39.2.3

Other properties

• The only injective object in Ring up to isomorphism is the zero ring (i.e. the terminal object). • Lacking zero morphisms, the category of rings cannot be a preadditive category. (However, every ring— considered as a small category with a single object— is a preadditive category). • The category of rings is a symmetric monoidal category with the tensor product of rings ⊗Z as the monoidal product and the ring of integers Z as the unit object. It follows from the Eckmann–Hilton theorem, that a monoid in Ring is just a commutative ring. The action of a monoid (= commutative ring) R on an object (= ring) A of Ring is just an R-algebra.

39.3. SUBCATEGORIES

103

39.3 Subcategories The category of rings has a number of important subcategories. These include the full subcategories of commutative rings, integral domains, principal ideal domains, and fields.

39.3.1

Category of commutative rings

The category of commutative rings, denoted CRing, is the full subcategory of Ring whose objects are all commutative rings. This category is one of the central objects of study in the subject of commutative algebra. Any ring can be made commutative by taking the quotient by the ideal generated by all elements of the form (xy − yx). This defines a functor Ring → CRing which is left adjoint to the inclusion functor, so that CRing is a reflective subcategory of Ring. The free commutative ring on a set of generators E is the polynomial ring Z[E] whose variables are taken from E. This gives a left adjoint functor to the forgetful functor from CRing to Set. CRing is limit-closed in Ring, which means that limits in CRing are the same as they are in Ring. Colimits, however, are generally different. They can be formed by taking the commutative quotient of colimits in Ring. The coproduct of two commutative rings is given by the tensor product of rings. Again, the coproduct of two nonzero commutative rings can be zero. The opposite category of CRing is equivalent to the category of affine schemes. The equivalence is given by the contravariant functor Spec which sends a commutative ring to its spectrum, an affine scheme.

39.3.2

Category of fields

The category of fields, denoted Field, is the full subcategory of CRing whose objects are fields. The category of fields is not nearly as well-behaved as other algebraic categories. In particular, free fields do not exist (i.e. there is no left adjoint to the forgetful functor Field → Set). It follows that Field is not a reflective subcategory of CRing. The category of fields is neither finitely complete nor finitely cocomplete. In particular, Field has neither products nor coproducts. Another curious aspect of the category of fields is that every morphism is a monomorphism. This follows from the fact that the only ideals in a field F are the zero ideal and F itself. One can then view morphisms in Field as field extensions. The category of fields is not connected. There are no morphisms between fields of different characteristic. The connected components of Field are the full subcategories of characteristic p, where p = 0 or is a prime number. Each such subcategory has an initial object: the prime field of characteristic p (which is Q if p = 0, otherwise the finite field Fp).

39.4 Related categories and functors 39.4.1

Category of groups

There is a natural functor from Ring to the category of groups, Grp, which sends each ring R to its group of units U(R) and each ring homomorphism to the restriction to U(R). This functor has a left adjoint which sends each group G to the integral group ring Z[G]. Another functor between these categories sends each ring R to the group of units of the matrix ring M2 (R) which acts on the projective line over a ring P(R).

39.4.2 R-algebras Given a commutative ring R one can define the category R-Alg whose objects are all R-algebras and whose morphisms are R-algebra homomorphisms. The category of rings can be considered a special case. Every ring can be considered a Z-algebra is a unique way. Ring homomorphisms are precisely the Z-algebra homomorphisms. The category of rings is, therefore, isomorphic

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to the category Z-Alg.[1] Many statements about the category of rings can be generalized to statements about the category of R-algebras. For each commutative ring R there is a functor R-Alg → Ring which forgets the R-module structure. This functor has a left adjoint which sends each ring A to the tensor product R⊗ZA, thought of as an R-algebra by setting r·(s⊗a) = rs⊗a.

39.4.3

Rings without identity

Many authors do not require rings to have a multiplicative identity element and, accordingly, do not require ring homomorphism to preserve the identity (should it exist). This leads to a rather different category. For distinction we call such algebraic structures rngs and their morphisms rng homomorphisms. The category of all rngs will be denoted by Rng. The category of rings, Ring, is a nonfull subcategory of Rng. Nonfull, because there are rng homomorphisms between rings which do not preserve the identity and are, therefore, not morphisms in Ring. The inclusion functor Ring → Rng has a left adjoint which formally adjoins an identity to any rng. This makes Ring into a (nonfull) reflective subcategory of Rng. The inclusion functor Ring → Rng respects limits but not colimits. The zero ring serves as both an initial and terminal object in Rng (that is, it is a zero object). It follows that Rng, like Grp but unlike Ring, has zero morphisms. These are just the rng homomorphisms that map everything to 0. Despite the existence of zero morphisms, Rng is still not a preadditive category. The pointwise sum of two rng homomorphisms is generally not a rng homomorphism. Coproducts in Rng are not the same as direct sums. There is a fully faithful functor from the category of abelian groups to Rng sending an abelian group to the associated rng of square zero. Free constructions are less natural in Rng then they are in Ring. For example, the free rng generated by a set {x} is the ring of all integral polynomials over x with no constant term, while the free ring generated by {x} is just the polynomial ring Z[x].

39.5 References [1] Tennison, B. R. (1975), Sheaf Theory, London Mathematical Society Lecture Note Series, Volume 20, Cambridge University Press, p. 74, ISBN 9780521207843.

• Adámek, Jiří; Horst Herrlich; George E. Strecker (1990). Abstract and Concrete Categories (PDF). John Wiley & Sons. ISBN 0-471-60922-6. • Mac Lane, Saunders; Garrett Birkhoff (1999). Algebra ((3rd ed.) ed.). Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-1646-2. • Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5 ((2nd ed.) ed.). Springer. ISBN 0-387-98403-8.

Chapter 40

Central polynomial In algebra, a central polynomial for n-by-n matrices is a polynomial in non-commuting variables that is non-constant but yields a scalar matrix whenever it is evaluated at n-by-n matrices. That such polynomials exist for any square matrices was discovered in 1970 independently by Formanek and Razmyslov. The term “central” is because the evaluation of a central polynomial has the image lying in the center of the matrix ring over any commutative ring. The notion has an application to the theory of polynomial identity rings. Example: (xy − yx)2 is a central polynomial for 2-by-2-matrices. Indeed, by the Cayley–Hamilton theorem, one has that (xy − yx)2 = − det(xy − yx)I for any 2-by-2-matrices x, y.

40.1 See also • Generic matrix ring

40.2 References • Formanek, Edward (1991). The polynomial identities and invariants of n×n matrices. Regional Conference Series in Mathematics 78. Providence, RI: American Mathematical Society. ISBN 0-8218-0730-7. Zbl 0714.16001. • Artin, Michael (1999). “Noncommutative Rings”. V. 4.

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Central simple algebra In ring theory and related areas of mathematics a central simple algebra (CSA) over a field K is a finite-dimensional associative algebra A, which is simple, and for which the center is exactly K. In other words, any simple algebra is a central simple algebra over its center. For example, the complex numbers C form a CSA over themselves, but not over the real numbers R (the center of C is all of C, not just R). The quaternions H form a 4-dimensional CSA over R, and in fact represent the only non-trivial element of the Brauer group of the reals (see below). Given two central simple algebras A ~ M(n,S) and B ~ M(m,T) over the same field F, A and B are called similar (or Brauer equivalent) if their division rings S and T are isomorphic. The set of all equivalence classes of central simple algebras over a given field F, under this equivalence relation, can be equipped with a group operation given by the tensor product of algebras. The resulting group is called the Brauer group Br(F) of the field F.[1] It is always a torsion group.[2]

41.1 Properties • According to the Artin–Wedderburn theorem a finite-dimensional simple algebra A is isomorphic to the matrix algebra M(n,S) for some division ring S. Hence, there is a unique division algebra in each Brauer equivalence class.[3] • Every automorphism of a central simple algebra is an inner automorphism (follows from Skolem–Noether theorem). • The dimension of a central simple algebra as a vector space over its centre is always a square: the degree is the square root of this dimension.[4] The Schur index of a central simple algebra is the degree of the equivalent division algebra:[5] it depends only on the Brauer class of the algebra.[6] • The period or exponent of a central simple algebra is the order of its Brauer class as an element of the Brauer group. It is a divisor of the index,[7] and the two numbers are composed of the same prime factors.[8][9][10] • If S is a simple subalgebra of a central simple algebra A then dimF S divides dimF A. • Every 4-dimensional central simple algebra over a field F is isomorphic to a quaternion algebra; in fact, it is either a two-by-two matrix algebra, or a division algebra. • If D is a central division algebra over K for which the index has prime factorisation

ind(D) =

r ∏

i pm i

i=1

then D has a tensor product decomposition D = ⊗ri=1 Di 106

41.2. SPLITTING FIELD

107

i where each component Di is a central division algebra of index pm , and the components are uniquely i [11] determined up to isomorphism.

41.2 Splitting field We call a field E a splitting field for A over K if A⊗E is isomorphic to a matrix ring over E. Every finite dimensional CSA has a splitting field: indeed, in the case when A is a division algebra, then a maximal subfield of A is a splitting field. In general by theorems of Wedderburn and Koethe there is a splitting field which is a separable extension of K of degree equal to the index of A, and this splitting field is isomorphic to a subfield of A.[12][13] As an example, the field C splits the quaternion algebra H over R with ( t + xi + yj + zk ↔

t + xi −y + zi

y + zi t − xi

) .

We can use the existence of the splitting field to define reduced norm and reduced trace for a CSA A.[14] Map A to a matrix ring over a splitting field and define the reduced norm and trace to be the composite of this map with determinant and trace respectively. For example, in the quaternion algebra H, the splitting above shows that the element t + x i + y j + z k has reduced norm t 2 + x2 + y2 + z2 and reduced trace 2t. The reduced norm is multiplicative and the reduced trace is additive. An element a of A is invertible if and only if its reduced norm in non-zero: hence a CSA is a division algebra if and only if the reduced norm is non-zero on the non-zero elements.[15]

41.3 Generalization CSAs over a field K are a non-commutative analog to extension fields over K – in both cases, they have no non-trivial 2-sided ideals, and have a distinguished field in their center, though a CSA can be non-commutative and need not have inverses (need not be a division algebra). This is of particular interest in noncommutative number theory as generalizations of number fields (extensions of the rationals Q); see noncommutative number field.

41.4 See also • Azumaya algebra, generalization of CSAs where the base field is replaced by a commutative local ring • Severi–Brauer variety • Posner’s theorem

41.5 References [1] Lorenz (2008) p.159 [2] Lorenz (2008) p.194 [3] Lorenz (2008) p.160 [4] Gille & Szamuely (2006) p.21 [5] Lorenz (2008) p.163 [6] Gille & Szamuely (2006) p.100 [7] Jacobson (1996) p.60 [8] Jacobson (1996) p.61 [9] Gille & Szamuely (2006) p.104

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[10] Cohn, Paul M. (2003). Further Algebra and Applications. Springer-Verlag. p. 208. ISBN 1852336676. [11] Gille & Szamuely (2006) p.105 [12] Jacobson (1996) pp.27-28 [13] Gille & Szamuely (2006) p.101 [14] Gille & Szamuely (2006) pp.37-38 [15] Gille & Szamuely (2006) p.38

• Cohn, P.M. (2003). Further Algebra and Applications (2nd ed.). Springer. ISBN 1852336676. Zbl 1006.00001. • Jacobson, Nathan (1996). Finite-dimensional division algebras over fields. Berlin: Springer-Verlag. ISBN 3-540-57029-2. Zbl 0874.16002. • Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023. • Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. ISBN 978-0-387-72487-4. Zbl 1130.12001.

41.5.1

Further reading

• Albert, A.A. (1939). Structure of Algebras. Colloquium Publications 24 (7th revised reprint ed.). American Mathematical Society. ISBN 0-8218-1024-3. Zbl 0023.19901. • Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001.

Chapter 42

Centralizer and normalizer “Normalizer” redirects here. For the process of increasing audio amplitude, see Audio normalization. “Centralizer” redirects here. For centralizers of Banach spaces, see Multipliers and centralizers (Banach spaces). In mathematics, especially group theory, the centralizer (also called commutant[1][2] ) of a subset S of a group G is the set of elements of G that commute with each element of S, and the normalizer of S is the set of elements of G that commute with S “as a whole”. The centralizer and normalizer of S are subgroups of G, and can provide insight into the structure of G. The definitions also apply to monoids and semigroups. In ring theory, the centralizer of a subset of a ring is defined with respect to the semigroup (multiplication) operation of the ring. The centralizer of a subset of a ring R is a subring of R. This article also deals with centralizers and normalizers in Lie algebra. The idealizer in a semigroup or ring is another construction that is in the same vein as the centralizer and normalizer.

42.1 Definitions Groups and semigroups The centralizer of a subset S of group (or semigroup) G is defined to be[3]

CG (S) = {g ∈ G | sg = gs all for s ∈ S} Sometimes if there is no ambiguity about the group in question, the G is suppressed from the notation entirely. When S={a} is a singleton set, then CG({a}) can be abbreviated to CG(a). Another less common notation for the centralizer is Z(a), which parallels the notation for the center of a group. With this latter notation, one must be careful to avoid confusion between the center of a group G, Z(G), and the centralizer of an element g in G, given by Z(g). The normalizer of S in the group (or semigroup) G is defined to be

NG (S) = {g ∈ G | gS = Sg} The definitions are similar but not identical. If g is in the centralizer of S and s is in S, then it must be that gs = sg, however if g is in the normalizer, gs = tg for some t in S, potentially different from s. The same conventions mentioned previously about suppressing G and suppressing braces from singleton sets also apply to the normalizer notation. The normalizer should not be confused with the normal closure. Rings, algebras, Lie rings and Lie algebras 109

110

CHAPTER 42. CENTRALIZER AND NORMALIZER

If R is a ring or an algebra, and S is a subset of the ring, then the centralizer of S is exactly as defined for groups, with R in the place of G. If L is a Lie algebra (or Lie ring) with Lie product [x,y], then the centralizer of a subset S of L is defined to be[4] CL (S) = {x ∈ L | [x, s] = 0 all for s ∈ S} The definition of centralizers for Lie rings is linked to the definition for rings in the following way. If R is an associative ring, then R can be given the bracket product [x,y] = xy − yx. Of course then xy = yx if and only if [x,y] = 0. If we denote the set R with the bracket product as LR, then clearly the ring centralizer of S in R is equal to the Lie ring centralizer of S in LR. The normalizer of a subset S of a Lie algebra (or Lie ring) L is given by[4] NL (S) = {x ∈ L | [x, s] ∈ S all for s ∈ S} While this is the standard usage of the term “normalizer” in Lie algebra, it should be noted that this construction is actually the idealizer of the set S in L . If S is an additive subgroup of L , then NL (S) is the largest Lie subring (or Lie subalgebra, as the case may be) in which S is a Lie ideal.[5]

42.2 Properties 42.2.1

Semigroups

Let S′ be the centralizer, i.e. S ′ = {x ∈ A : sx = xs for every s ∈ S}. Then: • S′ forms a subsemigroup. • S ′ = S ′′′ = S ′′′′′ - A commutant is its own bicommutant.

42.2.2 Source:

Groups [6]

• The centralizer and normalizer of S are both subgroups of G. • Clearly, CG(S)⊆NG(S). In fact, CG(S) is always a normal subgroup of NG(S). • CG(CG(S)) contains S, but CG(S) need not contain S. Containment will occur if st=ts for every s and t in S. Naturally then if H is an abelian subgroup of G, CG(H) contains H. • If S is a subsemigroup of G, then NG(S) contains S. • If H is a subgroup of G, then the largest subgroup in which H is normal is the subgroup NG(H). • A subgroup H of a group G is called a self-normalizing subgroup of G if NG(H) = H. • The center of G is exactly CG(G) and G is an abelian group if and only if CG(G)=Z(G) = G. • For singleton sets, CG(a)=NG(a). • By symmetry, if S and T are two subsets of G, T⊆CG(S) if and only if S⊆CG(T). • For a subgroup H of group G, the N/C theorem states that the factor group NG(H)/CG(H) is isomorphic to a subgroup of Aut(H), the automorphism group of H. Since NG(G) = G and CG(G) = Z(G), the N/C theorem also implies that G/Z(G) is isomorphic to Inn(G), the subgroup of Aut(G) consisting of all inner automorphisms of G. • If we define a group homomorphism T : G → Inn(G) by T(x)(g) = Tx(g) = xgx −1 , then we can describe NG(S) and CG(S) in terms of the group action of Inn(G) on G: the stabilizer of S in Inn(G) is T(NG(S)), and the subgroup of Inn(G) fixing S is T(CG(S)).

42.3. SEE ALSO

42.2.3 Source:

111

Rings and algebras [4]

• Centralizers in rings and algebras are subrings and subalgebras, respectively, and centralizers in Lie rings and Lie algebras are Lie subrings and Lie subalgebras, respectively. • The normalizer of S in a Lie ring contains the centralizer of S. • CR(CR(S)) contains S but is not necessarily equal. The double centralizer theorem deals with situations where equality occurs. • If S is an additive subgroup of a Lie ring A, then NA(S) is the largest Lie subring of A in which S is a Lie ideal. • If S is a Lie subring of a Lie ring A, then S⊆NA(S).

42.3 See also • Commutator • Stabilizer subgroup • Multipliers and centralizers (Banach spaces) • Double centralizer theorem • Idealizer

42.4 Notes [1] Kevin O'Meara; John Clark; Charles Vinsonhaler (2011). Advanced Topics in Linear Algebra: Weaving Matrix Problems Through the Weyr Form. Oxford University Press. p. 65. ISBN 978-0-19-979373-0. [2] Karl Heinrich Hofmann; Sidney A. Morris (2007). The Lie Theory of Connected Pro-Lie Groups: A Structure Theory for Pro-Lie Algebras, Pro-Lie Groups, and Connected Locally Compact Groups. European Mathematical Society. p. 30. ISBN 978-3-03719-032-6. [3] Jacobson (2009), p. 41 [4] Jacobson 1979, p.28. [5] Jacobson 1979, p.57. [6] Isaacs 2009, Chapters 1−3.

42.5 References • Isaacs, I. Martin (2009), Algebra: a graduate course, Graduate Studies in Mathematics 100 (reprint of the 1994 original ed.), Providence, RI: American Mathematical Society, pp. xii+516, ISBN 978-0-8218-4799-2, MR 2472787 • Jacobson, Nathan (2009), Basic algebra 1 (2 ed.), Dover, ISBN 978-0-486-47189-1 • Jacobson, Nathan (1979), Lie algebras (republication of the 1962 original ed.), New York: Dover Publications Inc., pp. ix+331, ISBN 0-486-63832-4, MR 559927

Chapter 43

Change of rings In algebra, given a ring homomorphism f : R → S , there are three ways to change the coefficient ring of a module; namely, for a right R-module M and a right S-module N, • f! M = M ⊗R S , the induced module. • f∗ M = HomR (S, M ) , the coinduced module. • f ∗ N = NR , the restriction of scalars. They are related as adjoint functors:

f! : ModR ⇆ ModS : f ∗ and

f ∗ : ModS ⇆ ModR : f∗ . This is related to Shapiro’s lemma.

43.1 See also • Six operations

43.2 References • J.P. May, Notes on Tor and Ext

112

Chapter 44

Characteristic (algebra) In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring’s multiplicative identity element (1) in a sum to get the additive identity element (0); the ring is said to have characteristic zero if this sum never reaches the additive identity. That is, char(R) is the smallest positive number n such that

1 + ··· + 1 = 0 | {z } nsummands

if such a number n exists, and 0 otherwise. The characteristic may also be taken to be the exponent of the ring’s additive group, that is, the smallest positive n such that

a + ··· + a = 0 | {z } nsummands

for every element a of the ring (again, if n exists; otherwise zero). Some authors do not include the multiplicative identity element in their requirements for a ring (see ring), and this definition is suitable for that convention; otherwise the two definitions are equivalent due to the distributive law in rings.

44.1 Other equivalent characterizations • The characteristic is the natural number n such that nZ is the kernel of a ring homomorphism from Z to R; • The characteristic is the natural number n such that R contains a subring isomorphic to the factor ring Z/nZ, which would be the image of that homomorphism. • When the non-negative integers {0, 1, 2, 3, . . . } are partially ordered by divisibility, then 1 is the smallest and 0 is the largest. Then the characteristic of a ring is the smallest value of n for which n · 1 = 0. If nothing “smaller” (in this ordering) than 0 will suffice, then the characteristic is 0. This is the right partial ordering because of such facts as that char A × B is the least common multiple of char A and char B, and that no ring homomorphism ƒ : A → B exists unless char B divides char A. • The characteristic of a ring R is n ∈ {0, 1, 2, 3, . . . } precisely if the statement ka = 0 for all a ∈ R implies n is a divisor of k. The requirements of ring homomorphisms are such that there can be only one homomorphism from the ring of integers to any ring; in the language of category theory, Z is an initial object of the category of rings. Again this follows the convention that a ring has a multiplicative identity element (which is preserved by ring homomorphisms). 113

114

CHAPTER 44. CHARACTERISTIC (ALGEBRA)

44.2 Case of rings If R and S are rings and there exists a ring homomorphism R → S, then the characteristic of S divides the characteristic of R. This can sometimes be used to exclude the possibility of certain ring homomorphisms. The only ring with characteristic 1 is the trivial ring which has only a single element 0 = 1. If a non-trivial ring R does not have any zero divisors, then its characteristic is either 0 or prime. In particular, this applies to all fields, to all integral domains, and to all division rings. Any ring of characteristic 0 is infinite. The ring Z/nZ of integers modulo n has characteristic n. If R is a subring of S, then R and S have the same characteristic. For instance, if q(X) is a prime polynomial with coefficients in the field Z/pZ where p is prime, then the factor ring (Z/pZ)[X]/(q(X)) is a field of characteristic p. Since the complex numbers contain the rationals, their characteristic is 0. A Z/nZ-algebra is equivalently a ring whose characteristic divides n. If a commutative ring R has prime characteristic p, then we have (x + y)p = xp + yp for all elements x and y in R – the "freshman’s dream" holds for power p. The map f(x) = xp then defines a ring homomorphism R → R. It is called the Frobenius homomorphism. If R is an integral domain it is injective.

44.3 Case of fields As mentioned above, the characteristic of any field is either 0 or a prime number. A field of non-zero characteristic is called a field of finite characteristic or a field of positive characteristic. For any field F, there is a minimal subfield, namely the prime field, the smallest subfield containing 1F. It is isomorphic either to the rational number field Q, or a finite field of prime order, Fp; the structure of the prime field and the characteristic each determine the other. Fields of characteristic zero have the most familiar properties; for practical purposes they resemble subfields of the complex numbers (unless they have very large cardinality, that is; in fact, any field of characteristic zero and cardinality at most continuum is isomorphic to a subfield of complex numbers).[1] The p-adic fields or any finite extension of them are characteristic zero fields, much applied in number theory, that are constructed from rings of characteristic pk , as k → ∞. For any ordered field, as the field of rational numbers Q or the field of real numbers R, the characteristic is 0. Thus, number fields and the field of complex numbers C are of characteristic zero. Actually, every field of characteristic zero is the quotient field of a ring Q[X]/P where X is a set of variables and P a set of polynomials in Q[X]. The finite field GF(pn ) has characteristic p. There exist infinite fields of prime characteristic. For example, the field of all rational functions over Z/pZ, the algebraic closure of Z/pZ or the field of formal Laurent series Z/pZ((T)). The characteristic exponent is defined similarly, except that it is equal to 1 if the characteristic is zero; otherwise it has the same value as the characteristic.[2] The size of any finite ring of prime characteristic p is a power of p. Since in that case it must contain Z/pZ it must also be a vector space over that field and from linear algebra we know that the sizes of finite vector spaces over finite fields are a power of the size of the field. This also shows that the size of any finite vector space is a prime power. (It is a vector space over a finite field, which we have shown to be of size pn . So its size is (pn )m = pnm .)

44.4 References [1] Enderton, Herbert B. (2001), A Mathematical Introduction to Logic (2nd ed.), Academic Press, p. 158, ISBN 9780080496467. Enderton states this result explicitly only for algebraically closed fields, but also describes a decomposition of any field as an algebraic extension of a transcendental extension of its prime field, from which the result follows immediately.

44.4. REFERENCES

[2] “Field Characteristic Exponent”. Wolfram Mathworld. Wolfram Research. Retrieved May 27, 2015.

• Neal H. McCoy (1964, 1973) The Theory of Rings, Chelsea Publishing, page 4.

115

Chapter 45

Classification of Clifford algebras In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the structures of finite-dimensional real and complex Clifford algebras have been completely classified. In each case, the Clifford algebra is algebra isomorphic to a full matrix ring over R, C, or H (the quaternions), or to a direct sum of two such algebras, though not in a canonical way. Below it is shown that distinct Clifford algebras may be algebra isomorphic, as is the case of Cℓ₂,₀(R) and Cℓ₁,₁(R) which are both isomorphic to the ring of two-by-two matrices over the real numbers.

45.1 Notation and conventions The Clifford product is the manifest ring product for the Clifford algebra, and all algebra homomorphisms in this article are with respect to this ring product. Other products defined within Clifford algebras, such as the exterior product, are not used here. This article uses the (+) sign convention for Clifford multiplication so that v 2 = Q(v) for all vectors v ∈ V, where Q is the quadratic form on the vector space V. We will denote the algebra of n×n matrices with entries in the division algebra K by Mn(K) or M(n,K). The direct sum of two such identical algebras will be denoted by Mn2 (K) = Mn(K) ⊕ Mn(K).

45.2 Bott periodicity Clifford algebras exhibit a 2-fold periodicity over the complex numbers and an 8-fold periodicity over the real numbers, which is related to the same periodicities for homotopy groups of the stable unitary group and stable orthogonal group, and is called Bott periodicity. The connection is explained by the geometric model of loop spaces approach to Bott periodicity: there 2-fold/8-fold periodic embeddings of the classical groups in each other (corresponding to isomorphism groups of Clifford algebras), and their successive quotients are symmetric spaces which are homotopy equivalent to the loop spaces of the unitary/orthogonal group.

45.3 Complex case The complex case is particularly simple: every nondegenerate quadratic form on a complex vector space is equivalent to the standard diagonal form Q(u) = u21 + u22 + · · · + u2n where n = dim V, so there is essentially only one Clifford algebra in each dimension. We will denote the Clifford algebra on Cn with the standard quadratic form by Cℓn(C). 116

45.4. REAL CASE

117

There are two separate cases to consider, according to whether n is even or odd. When n is even the algebra Cℓn(C) is central simple and so by the Artin-Wedderburn theorem is isomorphic to a matrix algebra over C. When n is odd, the center includes not only the scalars but the pseudoscalars (degree n elements) as well. We can always find a normalized pseudoscalar ω such that ω2 = 1. Define the operators

P± =

1 (1 ± ω). 2

These two operators form a complete set of orthogonal idempotents, and since they are central they give a decomposition of Cℓn(C) into a direct sum of two algebras − ± Cℓn (C) = Cℓ+ n (C) ⊕ Cℓn (C) where Cℓn (C) = P± Cℓn (C) .

The algebras Cℓn± (C) are just the positive and negative eigenspaces of ω and the P± are just the projection operators. Since ω is odd these algebras are mixed by α (the linear map on V defined by v ↦ −v):

∓ α(Cℓ± n (C)) = Cℓn (C)

and therefore isomorphic (since α is an automorphism). These two isomorphic algebras are each central simple and so, again, isomorphic to a matrix algebra over C. The sizes of the matrices can be determined from the fact that the dimension of Cℓn(C) is 2n . What we have then is the following table:

The even subalgebra of Cℓn(C) is (non-canonically) isomorphic to Cℓn₋₁(C). When n is even, the even subalgebra can be identified with the block diagonal matrices (when partitioned into 2×2 block matrix). When n is odd, the even subalgebra are those elements of M(2m ,C) ⊕ M(2m ,C) for which the two factors are identical. Picking either piece then gives an isomorphism with Cℓn₋₁(C) ≅ M(2m ,C).

45.4 Real case The real case is significantly more complicated, exhibiting a periodicity of 8 rather than 2, and there is a 2-parameter family of Clifford algebras.

45.4.1

Classification of quadratic form

Firstly, there are non-isomorphic quadratic forms of a given degree, classified by signature. Every nondegenerate quadratic form on a real vector space is equivalent to the standard diagonal form:

Q(u) = u21 + · · · + u2p − u2p+1 − · · · − u2p+q where n = p + q is the dimension of the vector space. The pair of integers (p, q) is called the signature of the quadratic form. The real vector space with this quadratic form is often denoted Rp,q . The Clifford algebra on Rp,q is denoted Cℓp,q(R). A standard orthonormal basis {ei} for Rp,q consists of n = p + q mutually orthogonal vectors, p of which have norm +1 and q of which have norm −1.

45.4.2

Unit pseudoscalar

See also: Pseudoscalar

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CHAPTER 45. CLASSIFICATION OF CLIFFORD ALGEBRAS

The unit pseudoscalar in Cℓp,q(R) is defined as ω = e1 e2 · · · en . This is both a Coxeter element of sorts (product of reflections) and a longest element of a Coxeter group in the Bruhat order; this is an analogy. It corresponds to and generalizes a volume form (in the exterior algebra; for the trivial quadratic form, the unit pseudoscalar is a volume form), and lifts reflection through the origin (meaning that the image of the unit pseudoscalar is reflection through the origin, in the orthogonal group). To compute the square ω 2 = (e1 e2 · · · en )(e1 e2 · · · en ) , one can either reverse the order of the second group, yielding sgn(σ)e1 e2 · · · en en · · · e2 e1 , or apply a perfect shuffle, yielding sgn(σ)(e1 e1 e2 e2 · · · en en ) . These both have sign (−1)⌊n/2⌋ = (−1)n(n−1)/2 , which is 4-periodic (proof), and combined with ei ei = ±1 , this shows that the square of ω is given by { 2

ω = (−1)

n(n−1)/2

q

(−1) = (−1)

(p−q)(p−q−1)/2

=

+1 p − q ≡ 0, 1 mod 4 −1 p − q ≡ 2, 3 mod 4.

Note that, unlike the complex case, it is not always possible to find a pseudoscalar which squares to +1.

45.4.3

Center

If n (equivalently, p − q) is even, the algebra Cℓp,q(R) is central simple and so isomorphic to a matrix algebra over R or H by the Artin–Wedderburn theorem. If n (equivalently, p − q) is odd then the algebra is no longer central simple but rather has a center which includes the pseudoscalars as well as the scalars. If n is odd and ω2 = +1 (equivalently, if p − q ≡ 1 (mod 4)) then, just as in the complex case, the algebra Cℓp,q(R) decomposes into a direct sum of isomorphic algebras − Cℓp,q (R) = Cℓ+ p,q (R) ⊕ Cℓp,q (R)

each of which is central simple and so isomorphic to matrix algebra over R or H. If n is odd and ω2 = −1 (equivalently, if p − q ≡ −1 (mod 4)) then the center of Cℓp,q(R) is isomorphic to C and can be considered as a complex algebra. As a complex algebra, it is central simple and so isomorphic to a matrix algebra over C.

45.4.4

Classification

All told there are three properties which determine the class of the algebra Cℓp,q(R): • signature mod 2: n is even/odd: central simple or not • signature mod 4: ω2 = ±1: if not central simple, center is R⊕R or C • signature mod 8: the Brauer class of the algebra (n even) or even subalgebra (n odd) is R or H Each of these properties depends only on the signature p − q modulo 8. The complete classification table is given below. The size of the matrices is determined by the requirement that Cℓp,q(R) have dimension 2p+q .

It may be seen that of all matrix ring types mentioned, there is only one type shared between both complex and real algebras: the type M(2m ,C). For example, Cℓ2 (C) and Cℓ₃,₀(R) are both determined to be M2 (C). It is important to note that there is a difference in the classifying isomorphisms used. Since the Cℓ2 (C) is algebra isomorphic via a C-linear map (which is necessarily R-linear), and Cℓ₃,₀(R) is algebra isomorphic via an R-linear map, Cℓ2 (C) and Cℓ₃,₀(R) are R-algebra isomorphic. A table of this classification for p + q ≤ 8 follows. Here p + q runs vertically and p − q runs horizontally (e.g. the algebra Cℓ₁,₃(R) ≅ M2 (H) is found in row 4, column −2).

45.5. BIBLIOGRAPHY

45.4.5

119

Symmetries

There is a tangled web of symmetries and relationships in the above table.

Cℓp+1,q+1 (R) = M2 (Cℓp,q (R)) Cℓp+4,q (R) = Cℓp,q+4 (R) Going over 4 spots in any row yields an identical algebra. From these Bott periodicity follows:

Cℓp+8,q (R) = Cℓp+4,q+4 (R) = M24 (Cℓp,q (R)). If the signature satisfies p − q ≡ 1 (mod 4) then

Cℓp+k,q (R) = Cℓp,q+k (R). (The table is symmetric about columns with signature 1, 5, −3, −7, and so forth.) Thus if the signature satisfies p − q ≡ 1 (mod 4),

Cℓp+k,q (R) = Cℓp,q+k (R) = Cℓp−k+k,q+k (R) = M2k (Cℓp−k,q (R)) = M2k (Cℓp,q−k (R)).

45.5 Bibliography • Paolo Budinich and Andrzej Trautman, The Spinorial Chessboard, Springer Verlag, Berlin-New York 1988, ISBN 9783540190783.

45.6 See also • Dirac algebra Cℓ₁,₃(C) • Pauli algebra Cℓ₃,₀(C) • Spacetime algebra Cℓ₁,₃(R) • Clifford module • Spin representation

Chapter 46

Clifford algebra This article is about (orthogonal) Clifford algebra. For symplectic Clifford algebra, see Weyl algebra. In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems.[1][2] The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English geometer William Kingdon Clifford. The most familiar Clifford algebra, or orthogonal Clifford algebra, is also referred to as Riemannian Clifford algebra.[3]

46.1 Introduction and basic properties A Clifford algebra is a unital associative algebra that contains and is generated by a vector space V over a field K, where V is equipped with a quadratic form Q. The Clifford algebra Cℓ(V, Q) is the “freest” algebra generated by V subject to the condition[4] v 2 = Q(v)1 all for v ∈ V, where the product on the left is that of the algebra, and the 1 is its multiplicative identity. The definition of a Clifford algebra endows the algebra with more structure than a “bare” K-algebra: specifically it has a designated or privileged subspace V. Such a subspace cannot in general be uniquely determined given only a K-algebra isomorphic to the Clifford algebra. If the characteristic of the ground field K is not 2, then one can rewrite this fundamental identity in the form uv + vu = 2⟨u, v⟩1 all for u, v ∈ V, where 1 (Q(u + v) − Q(u) − Q(v)) 2 is the symmetric bilinear form associated with Q, via the polarization identity. The idea of being the “freest” or “most general” algebra subject to this identity can be formally expressed through the notion of a universal property, as done below. ⟨u, v⟩ =

Quadratic forms and Clifford algebras in characteristic 2 form an exceptional case. In particular, if char(K) = 2 it is not true that a quadratic form determines a symmetric bilinear form, or that every quadratic form admits an orthogonal basis. Many of the statements in this article include the condition that the characteristic is not 2, and are false if this condition is removed. 120

46.2. UNIVERSAL PROPERTY AND CONSTRUCTION

46.1.1

121

As a quantization of the exterior algebra

Clifford algebras are closely related to exterior algebras. In fact, if Q = 0 then the Clifford algebra Cℓ(V, Q) is just the exterior algebra Λ(V). For nonzero Q there exists a canonical linear isomorphism between Λ(V) and Cℓ(V, Q) whenever the ground field K does not have characteristic two. That is, they are naturally isomorphic as vector spaces, but with different multiplications (in the case of characteristic two, they are still isomorphic as vector spaces, just not naturally). Clifford multiplication together with the privileged subspace is strictly richer than the exterior product since it makes use of the extra information provided by Q. More precisely, Clifford algebras may be thought of as quantizations (cf. Quantum group) of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra. Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras.

46.2 Universal property and construction Let V be a vector space over a field K, and let Q: V → K be a quadratic form on V. In most cases of interest the field K is either the field of real numbers R, or the field of complex numbers C, or a finite field. A Clifford algebra Cℓ(V, Q) is a unital associative algebra over K together with a linear map i : V → Cℓ(V, Q) satisfying i(v)2 = Q(v)1 for all v ∈ V, defined by the following universal property: given any associative algebra A over K and any linear map j : V → A such that j(v)2 = Q(v)1A for all v ∈ V (where 1A denotes the multiplicative identity of A), there is a unique algebra homomorphism f : Cℓ(V, Q) → A such that the following diagram commutes (i.e. such that f ∘ i = j):

Working with a symmetric bilinear form instead of Q (in characteristic not 2), the requirement on j is

j(v)j(w) + j(w)j(v) = 2⟨v, w⟩1A

for all v, w ∈ V .

A Clifford algebra as described above always exists and can be constructed as follows: start with the most general algebra that contains V, namely the tensor algebra T(V), and then enforce the fundamental identity by taking a suitable quotient. In our case we want to take the two-sided ideal IQ in T(V) generated by all elements of the form v ⊗ v − Q(v)1 for all v ∈ V and define Cℓ(V, Q) as the quotient algebra Cℓ(V, Q) = T(V)/IQ. The ring product inherited by this quotient is sometimes referred to as the Clifford product[5] to differentiate it from the exterior product and the scalar product. It is then straightforward to show that Cℓ(V, Q) contains V and satisfies the above universal property, so that Cℓ is unique up to a unique isomorphism; thus one speaks of “the” Clifford algebra Cℓ(V, Q). It also follows from this construction that i is injective. One usually drops the i and considers V as a linear subspace of Cℓ(V, Q).

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CHAPTER 46. CLIFFORD ALGEBRA

The universal characterization of the Clifford algebra shows that the construction of Cℓ(V, Q) is functorial in nature. Namely, Cℓ can be considered as a functor from the category of vector spaces with quadratic forms (whose morphisms are linear maps preserving the quadratic form) to the category of associative algebras. The universal property guarantees that linear maps between vector spaces (preserving the quadratic form) extend uniquely to algebra homomorphisms between the associated Clifford algebras.

46.3 Basis and dimension If the dimension of V over K is n and {e1 , …, en} is an orthogonal basis of (V, Q), then Cℓ(V, Q) is free over K with a basis

{ei1 ei2 · · · eik | 1 ≤ i1 < i2 < · · · < ik ≤ n and 0 ≤ k ≤ n} The empty product (k = 0) is defined as the multiplicative identity element. For each value of k there are n choose k basis elements, so the total dimension of the Clifford algebra is

dim Cℓ(V, Q) =

n ( ) ∑ n k=0

k

= 2n .

Since V comes equipped with a quadratic form, there is a set of privileged bases for V: the orthogonal ones. An orthogonal basis is one such that ⟨ei , ej ⟩ = 0 for i ̸= j , and ⟨ei , ei ⟩ = Q(ei ). where ⟨−, −⟩ is the symmetric bilinear form associated to Q. The fundamental Clifford identity implies that for an orthogonal basis ei ej = −ej ei for i ̸= j , and e2i = Q(ei ) This makes manipulation of orthogonal basis vectors quite simple. Given a product ei1 ei2 · · · eik of distinct orthogonal basis vectors of V, one can put them into standard order while including an overall sign determined by the number of pairwise swaps needed to do so (i.e. the signature of the ordering permutation).

46.4 Examples: real and complex Clifford algebras The most important Clifford algebras are those over real and complex vector spaces equipped with nondegenerate quadratic forms. It turns out that every one of the algebras Cℓp,q(R) and Cℓn(C) is isomorphic to A or A⊕A, where A is a full matrix ring with entries from R, C, or H. For a complete classification of these algebras see classification of Clifford algebras.

46.4.1

Real numbers

Main article: Geometric algebra The geometric interpretation of real Clifford algebras is known as geometric algebra. Every nondegenerate quadratic form on a finite-dimensional real vector space is equivalent to the standard diagonal form:

2 2 Q(v) = v12 + · · · + vp2 − vp+1 − · · · − vp+q

46.5. EXAMPLES: CONSTRUCTING QUATERNIONS AND DUAL QUATERNIONS

123

where n = p + q is the dimension of the vector space. The pair of integers (p, q) is called the signature of the quadratic form. The real vector space with this quadratic form is often denoted Rp, q . The Clifford algebra on Rp, q is denoted Cℓp, q(R). The symbol Cℓn(R) means either Cℓn,₀(R) or Cℓ₀,n(R) depending on whether the author prefers positive definite or negative definite spaces. A standard orthonormal basis {ei} for Rp,q consists of n = p + q mutually orthogonal vectors, p of which have norm +1 and q of which have norm −1. The algebra Cℓp,q(R) will therefore have p vectors that square to +1 and q vectors that square to −1. Note that Cℓ₀,₀(R) is naturally isomorphic to R since there are no nonzero vectors. Cℓ₀,₁(R) is a two-dimensional algebra generated by a single vector e1 that squares to −1, and therefore is isomorphic to C, the field of complex numbers. The algebra Cℓ₀,₂(R) is a four-dimensional algebra spanned by {1, e1 , e2 , e1 e2 }. The latter three elements square to −1 and all anticommute, and so the algebra is isomorphic to the quaternions H. Cℓ₀,₃(R) is an 8-dimensional algebra isomorphic to the direct sum H ⊕ H called split-biquaternions.

46.4.2

Complex numbers

One can also study Clifford algebras on complex vector spaces. Every nondegenerate quadratic form on a complex vector space is equivalent to the standard diagonal form

Q(z) = z12 + z22 + · · · + zn2 where n = dim V, up to isomorphism so there is only one nondegenerate Clifford algebra for each dimension n. We will denote the Clifford algebra on Cn with the standard quadratic form by Cℓn(C). The first few cases are not hard to compute. One finds that Cℓ0 (C) ≅ C, the complex numbers Cℓ1 (C) ≅ C ⊕ C, the bicomplex numbers Cℓ2 (C) ≅ M(2, C), the biquaternions where M(n, C) denotes the algebra of n×n matrices over C.

46.5 Examples: constructing quaternions and dual quaternions 46.5.1

Quaternions

In this section, Hamilton’s quaternions are constructed as the even sub algebra of the Clifford algebra Cℓ₀,₃(R). Let the vector space V be real three dimensional space R3 , and the quadratic form Q be derived from the usual Euclidean metric. Then, for v, w in R3 we have the quadratic form, or scalar product,

v · w = v1 w1 + v2 w2 + v3 w3 . Now introduce the Clifford product of vectors v and w given by

vw + wv = −2(v · w). This formulation uses the negative sign so the correspondence with quaternions is easily shown. Denote a set of orthogonal unit vectors of R3 as e1 , e2 , and e3 , then the Clifford product yields the relations

e2 e3 = −e3 e2 , e3 e1 = −e1 e3 , e1 e2 = −e2 e1 ,

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and

e21 = e22 = e23 = −1. The general element of the Clifford algebra Cℓ₀,₃(R) is given by

A = a0 + a1 e1 + a2 e2 + a3 e3 + a4 e2 e3 + a5 e3 e1 + a6 e1 e2 + a7 e1 e2 e3 . The linear combination of the even grade elements of Cℓ₀,₃(R) defines the even sub algebra Cℓ0 ₀,₃(R) with the general element

Q = q0 + q1 e2 e3 + q2 e3 e1 + q3 e1 e2 . The basis elements can be identified with the quaternion basis elements i, j, k as

i = e2 e3 , j = e3 e1 , k = e1 e2 , which shows that the even sub algebra Cℓ0 ₀,₃(R) is Hamilton’s real quaternion algebra. To see this, compute

i2 = (e2 e3 )2 = e2 e3 e2 e3 = −e2 e2 e3 e3 = −1, and

ij = e2 e3 e3 e1 = −e2 e1 = e1 e2 = k. Finally,

ijk = e2 e3 e3 e1 e1 e2 = −1.

46.5.2

Dual quaternions

In this section, dual quaternions are constructed as the even Clifford algebra of real four dimensional space with a degenerate quadratic form.[6][7] Let the vector space V be real four dimensional space R4 , and let the quadratic form Q be a degenerate form derived from the Euclidean metric on R3 . For v, w in R4 introduce the degenerate bilinear form

d(v, w) = v1 w1 + v2 w2 + v3 w3 . This degenerate scalar product projects distance measurements in R4 onto the R3 hyperplane. The Clifford product of vectors v and w is given by

vw + wv = −2 d(v, w). Note the negative sign is introduced to simplify the correspondence with quaternions. Denote a set of orthogonal unit vectors of R4 as e1 , e2 , e3 and e4 , then the Clifford product yields the relations

46.6. EXAMPLES: IN SMALL DIMENSION

125

em en = −en em , m ̸= n, and

e21 = e22 = e23 = −1, e24 = 0. The general element of the Clifford algebra Cℓ(R4 ,d) has 16 components. The linear combination of the even graded elements defines the even sub algebra Cℓ0 (R4 ,d) with the general element

H = h0 + h1 e2 e3 + h2 e3 e1 + h3 e1 e2 + h4 e4 e1 + h5 e4 e2 + h6 e4 e3 + h7 e1 e2 e3 e4 . The basis elements can be identified with the quaternion basis elements i, j, k and the dual unit ε as

i = e2 e3 , j = e3 e1 , k = e1 e2 , ε = e1 e2 e3 e4 . This provides the correspondence of Cℓ0 ₀,₃,₁(R) with dual quaternion algebra. To see this, compute

ε2 = (e1 e2 e3 e4 )2 = e1 e2 e3 e4 e1 e2 e3 e4 = −e1 e2 e3 (e4 e4 )e1 e2 e3 = 0, and

εi = (e1 e2 e3 e4 )e2 e3 = e1 e2 e3 e4 e2 e3 = e2 e3 (e1 e2 e3 e4 ) = iε. The exchanges of e1 and e4 alternate signs an even number of times, and show the dual unit ε commutes with the quaternion basis elements i, j, and k.

46.6 Examples: in small dimension Let K be any field of characteristic not 2.

46.6.1

Rank 1

If Q has diagonalization ⟨a⟩, that is there is a non-zero vector x such that Q(x)=a, then Cℓ(V, Q) is a K-algebra generated by an element x satisfiying x2 =a, so it is the étale quadratic algebra K[X]/(X-a). In particular, if a=0 (that is, Q is the zero quadratic form) then Cℓ(V, Q) is the dual numbers algebra over K. If a is a non-zero square in K, then Cℓ(V, Q)=K×K. Otherwise, Cℓ(V, Q) is the quadratic field extension K(√a) of K.

46.6.2

Rank 2

If Q has diagonalization ⟨a,b⟩ with non-zero a and b (which always exists if Q is non-degenerated), then Cℓ(V, Q) is a K-algebra generated by elements x and y satisfiying x2 =a, x2 =a and xy=-yx. Thus Cℓ(V, Q) is the (generalized) quaternion algebra (a,b)K. We retrieve the real case when a=b=−1, since H=(−1,−1)R. As a special case, if some x in V satisfies Q(x)=1, then Cℓ(V, Q) = M 2 (K).

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46.7 Properties 46.7.1

Relation to the exterior algebra

Given a vector space V one can construct the exterior algebra Λ(V), whose definition is independent of any quadratic form on V. It turns out that if K does not have characteristic 2 then there is a natural isomorphism between Λ(V) and Cℓ(V, Q) considered as vector spaces (and there exists an isomorphism in characteristic two, which may not be natural). This is an algebra isomorphism if and only if Q = 0. One can thus consider the Clifford algebra Cℓ(V, Q) as an enrichment (or more precisely, a quantization, cf. the Introduction) of the exterior algebra on V with a multiplication that depends on Q (one can still define the exterior product independent of Q). The easiest way to establish the isomorphism is to choose an orthogonal basis {ei} for V and extend it to a basis for Cℓ(V, Q) as described above. The map Cℓ(V, Q) → Λ(V) is determined by

ei1 ei2 · · · eik 7→ ei1 ∧ ei2 ∧ · · · ∧ eik . Note that this only works if the basis {ei} is orthogonal. One can show that this map is independent of the choice of orthogonal basis and so gives a natural isomorphism. If the characteristic of K is 0, one can also establish the isomorphism by antisymmetrizing. Define functions fk: V × … × V → Cℓ(V, Q) by

fk (v1 , · · · , vk ) =

1 ∑ sgn(σ) vσ(1) · · · vσ(k) k! σ∈Sk

where the sum is taken over the symmetric group on k elements. Since fk is alternating it induces a unique linear map Λk (V) → Cℓ(V, Q). The direct sum of these maps gives a linear map between Λ(V) and Cℓ(V, Q). This map can be shown to be a linear isomorphism, and it is natural. A more sophisticated way to view the relationship is to construct a filtration on Cℓ(V, Q). Recall that the tensor algebra T(V) has a natural filtration: F 0 ⊂ F 1 ⊂ F 2 ⊂ …, where Fk contains sums of tensors with order ≤ k. Projecting this down to the Clifford algebra gives a filtration on Cℓ(V, Q). The associated graded algebra

GrF Cℓ(V, Q) =

⊕

F k /F k−1

k

is naturally isomorphic to the exterior algebra Λ(V). Since the associated graded algebra of a filtered algebra is always isomorphic to the filtered algebra as filtered vector spaces (by choosing complements of Fk in F k+1 for all k), this provides an isomorphism (although not a natural one) in any characteristic, even two.

46.7.2

Grading

In the following, assume that the characteristic is not 2.[8] Clifford algebras are Z2 -graded algebras (also known as superalgebras). Indeed, the linear map on V defined by v ↦ −v (reflection through the origin) preserves the quadratic form Q and so by the universal property of Clifford algebras extends to an algebra automorphism α: Cℓ(V, Q) → Cℓ(V, Q). Since α is an involution (i.e. it squares to the identity) one can decompose Cℓ(V, Q) into positive and negative eigenspaces of α

Cℓ(V, Q) = Cℓ0 (V, Q) ⊕ Cℓ1 (V, Q) where Cℓi (V, Q) = {x ∈ Cℓ(V, Q) | α(x) = (−1)i x}. Since α is an automorphism it follows that

46.7. PROPERTIES

127

Cℓ i (V, Q)Cℓ j (V, Q) = Cℓ i+j (V, Q) where the superscripts are read modulo 2. This gives Cℓ(V, Q) the structure of a Z2 -graded algebra. The subspace Cℓ0 (V, Q) forms a subalgebra of Cℓ(V, Q), called the even subalgebra. The subspace Cℓ1 (V, Q) is called the odd part of Cℓ(V, Q) (it is not a subalgebra). This Z2 -grading plays an important role in the analysis and application of Clifford algebras. The automorphism α is called the main involution or grade involution. Elements that are pure in this Z2 -grading are simply said to be even or odd. Remark. In characteristic not 2 the underlying vector space of Cℓ(V, Q) inherits an N-grading and a Z-grading from the canonical isomorphism with the underlying vector space of the exterior algebra Λ(V).[9] It is important to note, however, that this is a vector space grading only. That is, Clifford multiplication does not respect the N-grading or Z-grading, only the Z2 -grading: for instance if Q(v) ≠ 0, then v ∈ Cℓ1 (V, Q), but v2 ∈ Cℓ0 (V, Q), not in Cℓ2 (V, Q). Happily, the gradings are related in the natural way: Z2 ≅N/2N≅ Z/2Z. Further, the Clifford algebra is Z-filtered: Cℓ≤i (V, Q) ⋅ Cℓ≤j (V, Q) ⊂ Cℓ≤i+j (V, Q). The degree of a Clifford number usually refers to the degree in the N-grading. The even subalgebra Cℓ0 (V, Q) of a Clifford algebra is itself isomorphic to a Clifford algebra.[10][11] If V is the orthogonal direct sum of a vector a of nonzero norm Q(a) and a subspace U, then Cℓ0 (V, Q) is isomorphic to Cℓ(U, −Q(a)Q), where −Q(a)Q is the form Q restricted to U and multiplied by −Q(a). In particular over the reals this implies that Cℓ0p,q (R) ∼ = Cℓp,q−1 (R) for q > 0, and 0 Cℓ (R) ∼ = Cℓq,p−1 (R) for p > 0. p,q

In the negative-definite case this gives an inclusion Cℓ₀,n₋₁(R) ⊂ Cℓ₀,n(R), which extends the sequence R ⊂ C ⊂ H ⊂ H⊕H ⊂ …; Likewise, in the complex case, one can show that the even subalgebra of Cℓn(C) is isomorphic to Cℓn₋₁(C).

46.7.3

Antiautomorphisms

In addition to the automorphism α, there are two antiautomorphisms which play an important role in the analysis of Clifford algebras. Recall that the tensor algebra T(V) comes with an antiautomorphism that reverses the order in all products:

v1 ⊗ v2 ⊗ · · · ⊗ vk 7→ vk ⊗ · · · ⊗ v2 ⊗ v1 . Since the ideal IQ is invariant under this reversal, this operation descends to an antiautomorphism of Cℓ(V, Q) called the transpose or reversal operation, denoted by xt . The transpose is an antiautomorphism: (xy)t = yt xt . The transpose operation makes no use of the Z2 -grading so we define a second antiautomorphism by composing α and the transpose. We call this operation Clifford conjugation denoted x ¯

x ¯ = α(xt ) = α(x)t . Of the two antiautomorphisms, the transpose is the more fundamental.[12] Note that all of these operations are involutions. One can show that they act as ±1 on elements which are pure in the Z-grading. In fact, all three operations depend only on the degree modulo 4. That is, if x is pure with degree k then

α(x) = ±x

xt = ±x

x ¯ = ±x

where the signs are given by the following table:

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46.7.4

CHAPTER 46. CLIFFORD ALGEBRA

Clifford scalar product

When the characteristic is not 2, the quadratic form Q on V can be extended to a quadratic form on all of Cℓ(V, Q) (which we also denoted by Q). A basis independent definition of one such extension is

Q(x) = ⟨xt x⟩ where ⟨a⟩ denotes the scalar part of a (the grade 0 part in the Z-grading). One can show that

Q(v1 v2 · · · vk ) = Q(v1 )Q(v2 ) · · · Q(vk ) where the vi are elements of V – this identity is not true for arbitrary elements of Cℓ(V, Q). The associated symmetric bilinear form on Cℓ(V, Q) is given by

⟨x, y⟩ = ⟨xt y⟩. One can check that this reduces to the original bilinear form when restricted to V. The bilinear form on all of Cℓ(V, Q) is nondegenerate if and only if it is nondegenerate on V. It is not hard to verify that the transpose is the adjoint of left/right Clifford multiplication with respect to this inner product. That is,

⟨ax, y⟩ = ⟨x, at y⟩, and

⟨xa, y⟩ = ⟨x, yat ⟩.

46.8 Structure of Clifford algebras In this section we assume that the vector space V is finite dimensional and that the bilinear form of Q is non-singular. A central simple algebra over K is a matrix algebra over a (finite dimensional) division algebra with center K. For example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions. • If V has even dimension then Cℓ(V, Q) is a central simple algebra over K. • If V has even dimension then Cℓ0 (V, Q) is a central simple algebra over a quadratic extension of K or a sum of two isomorphic central simple algebras over K. • If V has odd dimension then Cℓ(V, Q) is a central simple algebra over a quadratic extension of K or a sum of two isomorphic central simple algebras over K. • If V has odd dimension then Cℓ0 (V, Q) is a central simple algebra over K. The structure of Clifford algebras can be worked out explicitly using the following result. Suppose that U has even dimension and a non-singular bilinear form with discriminant d, and suppose that V is another vector space with a quadratic form. The Clifford algebra of U+V is isomorphic to the tensor product of the Clifford algebras of U and (−1)dim(U)/2 dV, which is the space V with its quadratic form multiplied by (−1)dim(U)/2 d. Over the reals, this implies in particular that

Cℓp+2,q (R) = M2 (R) ⊗ Cℓq,p (R)

46.9. CLIFFORD GROUP

129

Cℓp+1,q+1 (R) = M2 (R) ⊗ Cℓp,q (R) Cℓp,q+2 (R) = H ⊗ Cℓq,p (R). These formulas can be used to find the structure of all real Clifford algebras and all complex Clifford algebras; see the classification of Clifford algebras. Notably, the Morita equivalence class of a Clifford algebra (its representation theory: the equivalence class of the category of modules over it) depends only on the signature (p − q) mod 8. This is an algebraic form of Bott periodicity.

46.9 Clifford group The class of Clifford groups was discovered by Rudolf Lipschitz.[13] In this section we assume that V is finite dimensional and the quadratic form Q is nondegenerate. An action on the elements of a Clifford algebra by the group of its invertible elements may be defined in terms of a twisted conjugation: twisted conjugation by x maps y ↦ x y α(x)−1 , where α is the main involution defined above. The Clifford group Γ is defined to be the set of invertible elements x that stabilize vectors under this action, meaning that for all v in V we have:

xvα(x)−1 ∈ V. This formula also defines an action of the Clifford group on the vector space V that preserves the norm Q, and so gives a homomorphism from the Clifford group to the orthogonal group. The Clifford group contains all elements r of V of nonzero norm, and these act on V by the corresponding reflections that take v to v−2⟨v,r⟩r/Q(r) (In characteristic 2 these are called orthogonal transvections rather than reflections.) The Clifford group Γ is the disjoint union of two subsets Γ0 and Γ1 , where Γi is the subset of elements of degree i. The subset Γ0 is a subgroup of index 2 in Γ. If V is a finite dimensional real vector space with positive definite (or negative definite) quadratic form then the Clifford group maps onto the orthogonal group of V with respect to the form (by the Cartan–Dieudonné theorem) and the kernel consists of the nonzero elements of the field K. This leads to exact sequences

1 → K ∗ → Γ → OV (K) → 1, 1 → K ∗ → Γ0 → SOV (K) → 1. Over other fields or with indefinite forms, the map is not in general onto, and the failure is captured by the spinor norm.

46.9.1

Spinor norm

For more details on this topic, see Spinor_norm § Galois_cohomology_and_orthogonal_groups. In arbitrary characteristic, the spinor norm Q is defined on the Clifford group by

Q(x) = xt x. It is a homomorphism from the Clifford group to the group K* of non-zero elements of K. It coincides with the quadratic form Q of V when V is identified with a subspace of the Clifford algebra. Several authors define the spinor norm slightly differently, so that it differs from the one here by a factor of −1, 2, or −2 on Γ1 . The difference is not very important in characteristic other than 2. The nonzero elements of K have spinor norm in the group K*2 of squares of nonzero elements of the field K. So when V is finite dimensional and non-singular we get an induced map from the orthogonal group of V to the group

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K*/K*2 , also called the spinor norm. The spinor norm of the reflection about r⊥ , for any vector r, has image Q(r) in K*/K*2 , and this property uniquely defines it on the orthogonal group. This gives exact sequences:

1 → {±1} → PinV (K) → OV (K) → K ∗ /K ∗2 , 1 → {±1} → SpinV (K) → SOV (K) → K ∗ /K ∗2 . Note that in characteristic 2 the group {±1} has just one element. From the point of view of Galois cohomology of algebraic groups, the spinor norm is a connecting homomorphism on cohomology. Writing μ2 for the algebraic group of square roots of 1 (over a field of characteristic not 2 it is roughly the same as a two-element group with trivial Galois action), the short exact sequence

1 → µ2 → PinV → OV → 1 yields a long exact sequence on cohomology, which begins

1 → H 0 (µ2 ; K) → H 0 (PinV ; K) → H 0 (OV ; K) → H 1 (µ2 ; K). The 0th Galois cohomology group of an algebraic group with coefficients in K is just the group of K-valued points: H 0 (G; K) = G(K), and H 1 (μ2 ; K) ≅ K*/K*2 , which recovers the previous sequence

1 → {±1} → PinV (K) → OV (K) → K ∗ /K ∗2 , where the spinor norm is the connecting homomorphism H 0 (OV; K) → H 1 (μ2 ; K).

46.10 Spin and Pin groups For more details on this topic, see Spin group, Pin group and spinor. In this section we assume that V is finite dimensional and its bilinear form is non-singular. (If K has characteristic 2 this implies that the dimension of V is even.) The Pin group PinV(K) is the subgroup of the Clifford group Γ of elements of spinor norm ±1, and similarly the Spin group SpinV(K) is the subgroup of elements of Dickson invariant 0 in PinV(K). When the characteristic is not 2, these are the elements of determinant 1. The Spin group usually has index 2 in the Pin group. Recall from the previous section that there is a homomorphism from the Clifford group onto the orthogonal group. We define the special orthogonal group to be the image of Γ0 . If K does not have characteristic 2 this is just the group of elements of the orthogonal group of determinant 1. If K does have characteristic 2, then all elements of the orthogonal group have determinant 1, and the special orthogonal group is the set of elements of Dickson invariant 0. There is a homomorphism from the Pin group to the orthogonal group. The image consists of the elements of spinor norm 1 ∈ K*/K*2 . The kernel consists of the elements +1 and −1, and has order 2 unless K has characteristic 2. Similarly there is a homomorphism from the Spin group to the special orthogonal group of V. In the common case when V is a positive or negative definite space over the reals, the spin group maps onto the special orthogonal group, and is simply connected when V has dimension at least 3. Further the kernel of this homomorphism consists of 1 and −1. So in this case the spin group, Spin(n), is a double cover of SO(n). Please note, however, that the simple connectedness of the spin group is not true in general: if V is Rp,q for p and q both at least 2 then the spin group is not simply connected. In this case the algebraic group Spinp,q is simply connected as an algebraic group, even though its group of real valued points Spinp,q(R) is not simply connected. This is a rather subtle point, which completely confused the authors of at least one standard book about spin groups.

46.11. SPINORS

131

46.11 Spinors Clifford algebras Cℓp,q(C), with p+q=2n even, are matrix algebras which have a complex representation of dimension 2n . By restricting to the group Pinp,q(R) we get a complex representation of the Pin group of the same dimension, called the spin representation. If we restrict this to the spin group Spinp,q(R) then it splits as the sum of two half spin representations (or Weyl representations) of dimension 2n−1 . If p+q=2n+1 is odd then the Clifford algebra Cℓp,q(C) is a sum of two matrix algebras, each of which has a representation of dimension 2n , and these are also both representations of the Pin group Pinp,q(R). On restriction to the spin group Spinp,q(R) these become isomorphic, so the spin group has a complex spinor representation of dimension 2n . More generally, spinor groups and pin groups over any field have similar representations whose exact structure depends on the structure of the corresponding Clifford algebras: whenever a Clifford algebra has a factor that is a matrix algebra over some division algebra, we get a corresponding representation of the pin and spin groups over that division algebra. For examples over the reals see the article on spinors.

46.11.1

Real spinors

For more details on this topic, see spinor. To describe the real spin representations, one must know how the spin group sits inside its Clifford algebra. The Pin group, Pinp,q is the set of invertible elements in Cℓp, q which can be written as a product of unit vectors:

Pinp,q = {v1 v2 . . . vr | ∀i ∥vi ∥ = ±1}. Comparing with the above concrete realizations of the Clifford algebras, the Pin group corresponds to the products of arbitrarily many reflections: it is a cover of the full orthogonal group O(p, q). The Spin group consists of those elements of Pinp, q which are products of an even number of unit vectors. Thus by the Cartan-Dieudonné theorem Spin is a cover of the group of proper rotations SO(p,q). Let α : Cℓ → Cℓ be the automorphism which is given by the mapping v ↦ −v acting on pure vectors. Then in particular, Spinp,q is the subgroup of Pinp, q whose elements are fixed by α. Let

Cℓ0p,q = {x ∈ Cℓp,q | α(x) = x}. (These are precisely the elements of even degree in Cℓp, q.) Then the spin group lies within Cℓ0 p, q. The irreducible representations of Cℓp, q restrict to give representations of the pin group. Conversely, since the pin group is generated by unit vectors, all of its irreducible representation are induced in this manner. Thus the two representations coincide. For the same reasons, the irreducible representations of the spin coincide with the irreducible representations of Cℓ0 p, q To classify the pin representations, one need only appeal to the classification of Clifford algebras. To find the spin representations (which are representations of the even subalgebra), one can first make use of either of the isomorphisms (see above) Cℓ0 p,q ≈ Cℓp,q₋₁, for q > 0 Cℓ0 p,q ≈ Cℓq,p₋₁, for p > 0 and realize a spin representation in signature (p,q) as a pin representation in either signature (p,q−1) or (q,p−1).

46.12 Applications

132

46.12.1

CHAPTER 46. CLIFFORD ALGEBRA

Differential geometry

One of the principal applications of the exterior algebra is in differential geometry where it is used to define the bundle of differential forms on a smooth manifold. In the case of a (pseudo-)Riemannian manifold, the tangent spaces come equipped with a natural quadratic form induced by the metric. Thus, one can define a Clifford bundle in analogy with the exterior bundle. This has a number of important applications in Riemannian geometry. Perhaps more importantly is the link to a spin manifold, its associated spinor bundle and spinc manifolds.

46.12.2

Physics

Clifford algebras have numerous important applications in physics. Physicists usually consider a Clifford algebra to be an algebra with a basis generated by the matrices γ0 ,…,γ3 called Dirac matrices which have the property that

γi γj + γj γi = 2ηij where η is the matrix of a quadratic form of signature (1, 3) (or (3, 1) corresponding to the two equivalent choices of metric signature). These are exactly the defining relations for the Clifford algebra Cℓ₁,₃(R) up to an unimportant factor of 2, whose complexification is Cℓ₁,₃(R)C which, by the classification of Clifford algebras, is isomorphic to the algebra of 4 × 4 complex matrices Cℓ4 (C) ≈ M4(C). However, it is best to retain the notation Cℓ₁,₃(R)C, since any transformation that takes the bilinear form to the canonical form is not a Lorentz transformation of the underlying spacetime. The Clifford algebra of spacetime used in physics thus has more structure than Cℓ4 (C). It has in addition a set of preferred transformations – Lorentz transformations. Whether complexification is necessary to begin with depends in part on conventions used and in part on how much one wants to incorporate straightforwardly, but complexification is most often necessary in quantum mechanics where the spin representation of the Lie algebra so(1, 3) sitting inside the Clifford algebra conventionally requires a complex Clifford algebra. For reference, the spin Lie algebra is given by i σ µν = − [γ µ , γ ν ], 4 [σ µν , σ ρτ ] = i(η τ µ σ ρν + η ντ σ µρ − η ρµ σ τ ν − η νρ σ µτ ). This is in the (3, 1) convention, hence fits in Cℓ₃,₁(R)C.[14] The Dirac matrices were first written down by Paul Dirac when he was trying to write a relativistic first-order wave equation for the electron, and give an explicit isomorphism from the Clifford algebra to the algebra of complex matrices. The result was used to define the Dirac equation and introduce the Dirac operator. The entire Clifford algebra shows up in quantum field theory in the form of Dirac field bilinears. The use of Clifford algebras to describe quantum theory has been advanced among others by Mario Schönberg,[15] by David Hestenes in terms of geometric calculus, by David Bohm and Basil Hiley and co-workers in form of a hierarchy of Clifford algebras, and by Elio Conte et al.[16][17]

46.12.3

Computer vision

Recently, Clifford algebras have been applied in the problem of action recognition and classification in computer vision. Rodriguez et al.[18] propose a Clifford embedding to generalize traditional MACH filters to video (3D spatiotemporal volume), and vector-valued data such as optical flow. Vector-valued data is analyzed using the Clifford Fourier Transform. Based on these vectors action filters are synthesized in the Clifford Fourier domain and recognition of actions is performed using Clifford Correlation. The authors demonstrate the effectiveness of the Clifford embedding by recognizing actions typically performed in classic feature films and sports broadcast television.

46.13 See also • Algebra of physical space, APS

46.14. NOTES

133

• Classification of Clifford algebras • Clifford module • Gamma matrices • Exterior algebra • Generalized Clifford algebra • Geometric algebra • Spin group • Spinor • Paravector • Cayley–Dickson construction • spinor bundle • Dirac operator • Clifford analysis • spin structure • quaternion • octonion • complex spin structure • hypercomplex number • Higher-dimensional gamma matrices

46.14 Notes [1] W. K. Clifford, “Preliminary sketch of bi-quaternions, Proc. London Math. Soc. Vol. 4 (1873) pp. 381-395 [2] W. K. Clifford, Mathematical Papers, (ed. R. Tucker), London: Macmillan, 1882. [3] see for ex. Z. Oziewicz, Sz. Sitarczyk: Parallel treatment of Riemannian and symplectic Clifford algebras. In: Artibano Micali, Roger Boudet, Jacques Helmstetter (eds.): Clifford Algebras and their Applications in Mathematical Physics, Kluwer Academic Publishers, ISBN 0-7923-1623-1, 1992, p. 83 [4] Mathematicians who work with real Clifford algebras and prefer positive definite quadratic forms (especially those working in index theory) sometimes use a different choice of sign in the fundamental Clifford identity. That is, they take v2 = −Q(v). One must replace Q with −Q in going from one convention to the other. [5] Lounesto 2001, §1.8. [6] J. M. McCarthy, An Introduction to Theoretical Kinematics, pp. 62–5, MIT Press 1990. [7] O. Bottema and B. Roth, Theoretical Kinematics, North Holland Publ. Co., 1979 [8] Thus the group algebra K[Z/2] is semisimple and the Clifford algebra splits into eigenspaces of the main involution. [9] The Z-grading is obtained from the N grading by appending copies of the zero subspace indexed with the negative integers. [10] Technically, it does not have the full structure of a Clifford algebra without a designated vector subspace. [11] We are still assuming that the characteristic is not 2.

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[12] The opposite is true when using the alternate (−) sign convention for Clifford algebras: it is the conjugate which is more important. In general, the meanings of conjugation and transpose are interchanged when passing from one sign convention to the other. For example, in the convention used here the inverse of a vector is given by v−1 = vt / Q(v) while in the (−) convention it is given by v−1 = v / Q(v). [13] Lounesto 2001, §17.2. [14] Weinberg 2002 [15] See the references to Schönberg’s papers of 1956 and 1957 as described in section “The Grassmann–Schönberg algebra Gn " of:A. O. Bolivar, Classical limit of fermions in phase space, J. Math. Phys. 42, 4020 (2001) doi:10.1063/1.1386411 [16] Conte, Elio (2002). “A Quantum Like Interpretation and Solution of Einstein, Podolsky, and Rosen Paradox in Quantum Mechanics”. arXiv:0711.2260 [quant-ph]. [17] Elio Conte: On some considerations of mathematical physics: May we identify Clifford algebra as a common algebraic structure for classical diffusion and Schrödinger equations? Adv. Studies Theor. Phys., vol. 6, no. 26 (2012), pp. 1289– 1307 [18] Rodriguez, Mikel; Shah, M (2008). “Action MACH: A Spatio-Temporal Maximum Average Correlation Height Filter for Action Classification”. Computer Vision and Pattern Recognition (CVPR).

46.15 References • Bourbaki, Nicolas (1988), Algebra, Berlin, New York: Springer-Verlag, ISBN 978-3-540-19373-9, section IX.9. • Carnahan, S. Borcherds Seminar Notes, Uncut. Week 5, “Spinors and Clifford Algebras”. • Garling, D. J. H. (2011), Clifford algebras. An introduction, London Mathematical Society Student Texts 78, Cambridge: Cambridge University Press, ISBN 978-1-107-09638-7, Zbl 1235.15025 • Lam, Tsit-Yuen (2005), Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67, American Mathematical Society, ISBN 0-8218-1095-2, MR 2104929, Zbl 1068.11023 • Lawson, H. Blaine; Michelsohn, Marie-Louise (1989), Spin Geometry, Princeton, NJ: Princeton University Press, ISBN 978-0-691-08542-5. An advanced textbook on Clifford algebras and their applications to differential geometry. • Lounesto, Pertti (2001), Clifford algebras and spinors, Cambridge: Cambridge University Press, ISBN 978-0521-00551-7 • Porteous, Ian R. (1995), Clifford algebras and the classical groups, Cambridge: Cambridge University Press, ISBN 978-0-521-55177-9 • Jagannathan, R., On generalized Clifford algebras and their physical applications, arXiv:1005.4300 • Sylvester, J. J., (1882), Johns Hopkins University Circulars I: 241-242; ibid II (1883) 46; ibid III (1884) 79. Summarized in The Collected Mathematics Papers of James Joseph Sylvester (Cambridge University Press, 1909) v III .online and further. • Weinberg, S. (2002), The Quantum Theory of Fields 1, Cambridge University Press, ISBN 0-521-55001-7

46.16 Further reading • Knus, Max-Albert (1991), Quadratic and Hermitian forms over rings, Grundlehren der Mathematischen Wissenschaften 294, Berlin: Springer-Verlag, doi:10.1007/978-3-642-75401-2, ISBN 3-540-52117-8, MR 1096299, Zbl 0756.11008

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46.17 External links • Hazewinkel, Michiel, ed. (2001), “Clifford algebra”, Encyclopedia of Mathematics, Springer, ISBN 978-155608-010-4 • Planetmath entry on Clifford algebras • A history of Clifford algebras (unverified) • John Baez on Clifford algebras • Clifford Algebra: A Visual Introduction

Chapter 47

Coherent ring In mathematics, a (left) coherent ring is a ring in which every finitely generated left ideal is finitely presented. Many theorems about finitely generated modules over Noetherian rings can be extended to finitely presented modules over coherent rings. Every left Noetherian ring is left-coherent. The ring of polynomials in an infinite number of variables is an example of a left-coherent ring that is not left Noetherian. A ring is left coherent if and only if every direct product of flat right modules is flat (Chase 1960), (Anderson & Fuller 1992, p. 229). Compare this to: A ring is left Noetherian if and only if every direct sum of injective left modules is injective.

47.1 References • Anderson, Frank Wylie; Fuller, Kent R (1992), Rings and Categories of Modules, Berlin, New York: SpringerVerlag, ISBN 978-0-387-97845-1 • Chase, Stephen U. (1960), “Direct products of modules”, Transactions of the American Mathematical Society (American Mathematical Society) 97 (3): 457–473, doi:10.2307/1993382, JSTOR 1993382, MR 0120260 • Govorov, V.E. (2001), “Coherent_ring”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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Commutative ring In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Some specific kinds of commutative rings are given with the following chain of class inclusions: Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields

48.1 Definition and first examples 48.1.1

Definition

For more details on the definition of rings, see Ring (mathematics). A ring is a set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called addition and multiplication and commonly denoted by "+" and "⋅"; e.g. a + b and a ⋅ b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under multiplication, where multiplication distributes over addition; i.e., a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c). The identity elements for addition and multiplication are denoted 0 and 1, respectively. If the multiplication is commutative, i.e. a ⋅ b = b ⋅ a, then the ring R is called commutative. In the remainder of this article, all rings will be commutative, unless explicitly stated otherwise.

48.1.2

First examples

An important example, and in some sense crucial, is the ring of integers Z with the two operations of addition and multiplication. As the multiplication of integers is a commutative operation, this is a commutative ring. It is usually denoted Z as an abbreviation of the German word Zahlen (numbers). A field is a commutative ring where every non-zero element a is invertible; i.e., has a multiplicative inverse b such that a ⋅ b = 1. Therefore, by definition, any field is a commutative ring. The rational, real and complex numbers form fields. The ring of 2×2 matrices is not commutative, since matrix multiplication fails to be commutative, as the following example shows: 137

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[

] [ 1 1 1 · 0 1 1 [ ] [ 1 1 1 · 1 0 0

] [ 1 2 = 0 1 ] [ 1 1 = 1 1

1 0 2 1

] ]

However, matrices that can be diagonalized with the same similarity transformation do form a commutative ring. An example is the set of matrices of divided differences with respect to a fixed set of nodes. If R is a given commutative ring, then the set of all polynomials in the variable X whose coefficients are in R forms the polynomial ring, denoted R[X]. The same holds true for several variables. If V is some topological space, for example a subset of some Rn , real- or complex-valued continuous functions on V form a commutative ring. The same is true for differentiable or holomorphic functions, when the two concepts are defined, such as for V a complex manifold.

48.2 Ideals and the spectrum In the following, R denotes a commutative ring. In contrast to fields, where every nonzero element is multiplicatively invertible, the theory of rings is more complicated. There are several notions to cope with that situation. First, an element a of ring R is called a unit if it possesses a multiplicative inverse. Another particular type of element is the zero divisors, i.e. a non-zero element a such that there exists a non-zero element b of the ring such that ab = 0. If R possesses no zero divisors, it is called an integral domain since it closely resembles the integers in some ways. Many of the following notions also exist for not necessarily commutative rings, but the definitions and properties are usually more complicated. For example, all ideals in a commutative ring are automatically two-sided, which simplifies the situation considerably.

48.2.1

Ideals and factor rings

Main articles: Ideal and Factor ring The inner structure of a commutative ring is determined by considering its ideals, i.e. nonempty subsets that are closed under multiplication with arbitrary ring elements and addition: for all r in R, i and j in I, both ri and i + j are required to be in I. Given any subset F = {fj}j ∈ J of R (where J is some index set), the ideal generated by F is the smallest ideal that contains F. Equivalently, it is given by finite linear combinations r1 f 1 + r2 f 2 + ... + rnfn. An ideal generated by one element is called a principal ideal. A ring all of whose ideals are principal is called a principal ideal ring; two important cases are Z and k[X], the polynomial ring over a field k. Any ring has two ideals, namely the zero ideal {0} and R, the whole ring. An ideal is proper if it is strictly smaller than the whole ring. An ideal that is not strictly contained in any proper ideal is called maximal. An ideal m is maximal if and only if R / m is a field. Except for the zero ring, any ring (with identity) possesses at least one maximal ideal; this follows from Zorn’s lemma. The definition of ideals is such that “dividing” I “out” gives another ring, the factor ring R / I: it is the set of cosets of I together with the operations (a + I) + (b + I) = (a + b) + I and (a + I)(b + I) = ab + I. For example, the ring Z/nZ (also denoted Zn), where n is an integer, is the ring of integers modulo n. It is the basis of modular arithmetic.

48.2. IDEALS AND THE SPECTRUM

48.2.2

139

Localizations

Main article: Localization of a ring The localization of a ring is the counterpart to factor rings insofar as in a factor ring R / I certain elements (namely the elements of I) become zero, whereas in the localization certain elements are rendered invertible, i.e. multiplicative inverses are added to the ring. Concretely, if S is a multiplicatively closed subset of R (i.e. whenever s, t ∈ S then so is st) then the localization of R at S, or ring of fractions with denominators in S, usually denoted S −1 R consists of symbols r s

with r ∈ R, s ∈ S

subject to certain rules that mimick the cancellation familiar from rational numbers. Indeed, in this language Q is the localization of Z at all nonzero integers. This construction works for any integral domain R instead of Z. The localization (R \ {0})−1 R is called the quotient field of R. If S consists of the powers of one fixed element f, the localisation is written Rf.

48.2.3

Prime ideals and the spectrum

Main articles: Prime ideal and Spectrum of a ring A particularly important type of ideals is prime ideals, often denoted p. This notion arose when algebraists (in the 19th century) realized that, unlike in Z, in many rings there is no unique factorization into prime numbers. (Rings where it does hold are called unique factorization domains.) By definition, a prime ideal is a proper ideal such that, whenever the product ab of any two ring elements a and b is in p, at least one of the two elements is already in p. (The opposite conclusion holds for any ideal, by definition). Equivalently, the factor ring R / p is an integral domain. Yet another way of expressing the same is to say that the complement R \ p is multiplicatively closed. The localisation (R \ p)−1 R is important enough to have its own notation: Rp. This ring has only one maximal ideal, namely pRp. Such rings are called local. By the above, any maximal ideal is prime. Proving that an ideal is prime, or equivalently that a ring has no zerodivisors can be very difficult.

The spectrum of Z.

Prime ideals are the key step in interpreting a ring geometrically, via the spectrum of a ring Spec R: it is the set of all prime ideals of R.[nb 1] As noted above, there is at least one prime ideal, therefore the spectrum is nonempty. If R is a field, the only prime ideal is the zero ideal, therefore the spectrum is just one point. The spectrum of Z, however, contains one point for the zero ideal, and a point for any prime number p (which generates the prime ideal pZ). The spectrum is endowed with a topology called the Zariski topology, which is determined by specifying that subsets D(f) = {p ∈ Spec R, f ∉ p}, where f is any ring element, be open. This topology tends to be different from those encountered in analysis or differential geometry; for example, there will generally be points which are not closed. The closure of the point corresponding to the zero ideal 0 ⊂ Z, for example, is the whole spectrum of Z. The notion of a spectrum is the common basis of commutative algebra and algebraic geometry. Algebraic geometry proceeds by endowing Spec R with a sheaf O (an entity that collects functions defined locally, i.e. on varying open subsets). The datum of the space and the sheaf is called an affine scheme. Given an affine scheme, the underlying ring R can be recovered as the global sections of O . Moreover, the established one-to-one correspondence between rings and affine schemes is also compatible with ring homomorphisms: any f : R → S gives rise to a continuous map in the opposite direction

140

CHAPTER 48. COMMUTATIVE RING Spec S → Spec R, q ↦ f −1 (q), i.e. any prime ideal of S is mapped to its preimage under f, which is a prime ideal of R.

The spectrum also makes precise the intuition that localisation and factor rings are complementary: the natural maps R → Rf and R → R / fR correspond, after endowing the spectra of the rings in question with their Zariski topology, to complementary open and closed immersions respectively. Altogether the equivalence of the two said categories is very apt to reflect algebraic properties of rings in a geometrical manner. Affine schemes are–much the same way as manifolds are locally given by open subsets of Rn –local models for schemes, which are the object of study in algebraic geometry. Therefore, many notions that apply to rings and homomorphisms stem from geometric intuition.

48.3 Ring homomorphisms Main article: Ring homomorphism As usual in algebra, a function f between two objects that respects the structures of the objects in question is called homomorphism. In the case of rings, a ring homomorphism is a map f : R → S such that f(a + b) = f(a) + f(b), f(ab) = f(a)f(b) and f(1) = 1. These conditions ensure f(0) = 0, but the requirement that the multiplicative identity element 1 is preserved under f would not follow from the two remaining properties. In such a situation S is also called an R-algebra, by understanding that s in S may be multiplied by some r of R, by setting r · s := f(r) · s. The kernel and image of f are defined by ker (f) = {r ∈ R, f(r) = 0} and im (f) = f(R) = {f(r), r ∈ R}. The kernel is an ideal of R, and the image is a subring of S.

48.4 Modules Main article: Modules The outer structure of a commutative ring is determined by considering linear algebra over that ring, i.e., by investigating the theory of its modules, which are similar to vector spaces, except that the base is not necessarily a field, but can be any ring R. The theory of R-modules is significantly more difficult than linear algebra of vector spaces. Module theory has to grapple with difficulties such as modules not having bases, that the rank of a free module (i.e. the analog of the dimension of vector spaces) may not be well-defined and that submodules of finitely generated modules need not be finitely generated (unless R is Noetherian, see below). Ideals within a ring R can be characterized as R-modules which are submodules of R. On the one hand, a good understanding of R-modules necessitates enough information about R. Vice versa, however, many techniques in commutative algebra that study the structure of R, by examining its ideals, proceed by studying modules in general.

48.5 Noetherian rings Main article: Noetherian ring A ring is called Noetherian (in honor of Emmy Noether, who developed this concept) if every ascending chain of ideals 0 ⊆ I 0 ⊆ I 1 ... ⊆ In ⊆ In ₊ ₁ ⊆ ...

48.6. DIMENSION

141

becomes stationary, i.e. becomes constant beyond some index n. Equivalently, any ideal is generated by finitely many elements, or, yet equivalent, submodules of finitely generated modules are finitely generated. A ring is called Artinian (after Emil Artin), if every descending chain of ideals R ⊇ I 0 ⊇ I 1 ... ⊇ In ⊇ In ₊ ₁ ⊇ ... becomes stationary eventually. Despite the two conditions appearing symmetric, Noetherian rings are much more general than Artinian rings. For example, Z is Noetherian, since every ideal can be generated by one element, but is not Artinian, as the chain Z ⊋ 2Z ⊋ 4Z ⊋ 8Z ⊋ ... shows. In fact, by the Hopkins–Levitzki theorem, every Artinian ring is Noetherian. Being Noetherian is an extremely important finiteness condition. The condition is preserved under many operations that occur frequently in geometry: if R is Noetherian, then so is the polynomial ring R[X1 , X2 , ..., Xn] (by Hilbert’s basis theorem), any localization S −1 R, factor rings R / I.

48.6 Dimension Main article: Krull dimension The Krull dimension (or simply dimension) dim R of a ring R is a notion to measure the “size” of a ring, very roughly by the counting independent elements in R. Precisely, it is defined as the supremum of lengths n of chains of prime ideals

p0 ⊊ p1 ⊊ . . . ⊊ pn For example, a field is zero-dimensional, since the only prime ideal is the zero ideal. It is also known that a commutative ring is Artinian if and only if it is Noetherian and zero-dimensional (i.e., all its prime ideals are maximal). The integers are one-dimensional: any chain of prime ideals is of the form 0 = p0 ⊊ pZ = p1 , where p is a prime number since any ideal in Z is principal. The dimension behaves well if the rings in question are Noetherian: the expected equality dim R[X] = dim R + 1 holds in this case (in general, one has only dim R + 1 ≤ dim R[X] ≤ 2 · dim R + 1). Furthermore, since the dimension depends only on one maximal chain, the dimension of R is the supremum of all dimensions of its localisations Rp, where p is an arbitrary prime ideal. Intuitively, the dimension of R is a local property of the spectrum of R. Therefore, the dimension is often considered for local rings only, also since general Noetherian rings may still be infinite, despite all their localisations being finite-dimensional. Determining the dimension of, say, k[X1 , X2 , ..., Xn] / (f 1 , f 2 , ..., fm), where k is a field and the fi are some polynomials in n variables, is generally not easy. For R Noetherian, the dimension of R / I is, by Krull’s principal ideal theorem, at least dim R − n, if I is generated by n elements. If the dimension does drops as much as possible, i.e. dim R / I = dim R − n, the R / I is called a complete intersection. A local ring R, i.e. one with only one maximal ideal m, is called regular, if the (Krull) dimension of R equals the dimension (as a vector space over the field R / m) of the cotangent space m / m2 .

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48.7 Constructing commutative rings There are several ways to construct new rings out of given ones. The aim of such constructions is often to improve certain properties of the ring so as to make it more readily understandable. For example, an integral domain that is integrally closed in its field of fractions is called normal. This is a desirable property, for example any normal one-dimensional ring is necessarily regular. Rendering a ring normal is known as normalization.

48.7.1

Completions

If I is an ideal in a commutative ring R, the powers of I form topological neighborhoods of 0 which allow R to be viewed as a topological ring. This topology is called the I-adic topology. R can then be completed with respect to this topology. Formally, the I-adic completion is the inverse limit of the rings R/In . For example, if k is a field, k[[X]], the formal power series ring in one variable over k, is the I-adic completion of k[X] where I is the principal ideal generated by X. Analogously, the ring of p-adic integers is the I-adic completion of Z where I is the principal ideal generated by p. Any ring that is isomorphic to its own completion, is called complete.

48.8 Properties By Wedderburn’s theorem, every finite division ring is commutative, and therefore a finite field. Another condition ensuring commutativity of a ring, due to Jacobson, is the following: for every element r of R there exists an integer n > 1 such that rn = r.[1] If, r2 = r for every r, the ring is called Boolean ring. More general conditions which guarantee commutativity of a ring are also known.[2]

48.9 See also • Graded ring • Almost commutative ring • Almost ring, a certain generalization of a commutative ring. • Simplicial commutative ring, a simplicial object in the category of commutative rings.

48.10 Notes [1] This notion can be related to the spectrum of a linear operator, see Spectrum of a C*-algebra and Gelfand representation.

48.10.1

Citations

[1] Jacobson 1945 [2] Pinter-Lucke 2007

48.11 References • Atiyah, Michael; Macdonald, I. G. (1969), Introduction to commutative algebra, Addison-Wesley Publishing Co. • Balcerzyk, Stanisław; Józefiak, Tadeusz (1989), Commutative Noetherian and Krull rings, Ellis Horwood Series: Mathematics and its Applications, Chichester: Ellis Horwood Ltd., ISBN 978-0-13-155615-7 • Balcerzyk, Stanisław; Józefiak, Tadeusz (1989), Dimension, multiplicity and homological methods, Ellis Horwood Series: Mathematics and its Applications., Chichester: Ellis Horwood Ltd., ISBN 978-0-13-155623-2

48.11. REFERENCES

143

• Eisenbud, David (1995), Commutative algebra. With a view toward algebraic geometry., Graduate Texts in Mathematics 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94268-1, MR 1322960 • Jacobson, Nathan (1945), “Structure theory of algebraic algebras of bounded degree”, Annals of Mathematics 46 (4): 695–707, doi:10.2307/1969205, ISSN 0003-486X, JSTOR 1969205 • Kaplansky, Irving (1974), Commutative rings (Revised ed.), University of Chicago Press, MR 0345945 • Matsumura, Hideyuki (1989), Commutative Ring Theory, Cambridge Studies in Advanced Mathematics (2nd ed.), Cambridge University Press, ISBN 978-0-521-36764-6 • Nagata, Masayoshi (1975) [1962], Local rings, Interscience Tracts in Pure and Applied Mathematics 13, Interscience Publishers, pp. xiii+234, ISBN 978-0-88275-228-0, MR 0155856 • Pinter-Lucke, James (2007), “Commutativity conditions for rings: 1950–2005”, Expositiones Mathematicae 25 (2): 165–174, doi:10.1016/j.exmath.2006.07.001, ISSN 0723-0869 • Zariski, Oscar; Samuel, Pierre (1958–60), Commutative Algebra I, II, University series in Higher Mathematics, Princeton, N.J.: D. van Nostrand, Inc. (Reprinted 1975-76 by Springer as volumes 28-29 of Graduate Texts in Mathematics.)

Chapter 49

Comodule In mathematics, a comodule or corepresentation is a concept dual to a module. The definition of a comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebra.

49.1 Formal definition Let K be a field, and C be a coalgebra over K. A (right) comodule over C is a K-vector space M together with a linear map

ρ:M →M ⊗C such that 1. (id ⊗ ∆) ◦ ρ = (ρ ⊗ id) ◦ ρ 2. (id ⊗ ε) ◦ ρ = id , where Δ is the comultiplication for C, and ε is the counit. Note that in the second rule we have identified M ⊗ K with M .

49.2 Examples • A coalgebra is a comodule over itself. • If M is a finite-dimensional module over a finite-dimensional K-algebra A, then the set of linear functions from A to K forms a coalgebra, and the set of linear functions from M to K forms a comodule over that coalgebra. • A graded vector space V can be made into a comodule. Let I be the index set for the graded vector space, and let CI be the vector space with basis ei for i ∈ I . We turn CI into a coalgebra and V into a CI -comodule, as follows: 1. Let the comultiplication on CI be given by ∆(ei ) = ei ⊗ ei . 2. Let the counit on CI be given by ε(ei ) = 1 . ∑ 3. Let the map ρ on V be given by ρ(v) = vi ⊗ ei , where vi is the i-th homogeneous piece of v . 144

49.3. RATIONAL COMODULE

145

49.3 Rational comodule If M is a (right) comodule over the coalgebra C, then M is a (left) module over the dual algebra C ∗ , but the converse is not true in general: a module over C ∗ is not necessarily a comodule over C. A rational comodule is a module over C ∗ which becomes a comodule over C in the natural way.

49.4 References Montgomery, Susan (1993). Hopf algebras and their actions on rings. Regional Conference Series in Mathematics 82. Providence, RI: American Mathematical Society. ISBN 0-8218-0738-2. Zbl 0793.16029.

Chapter 50

Comparison of vector algebra and geometric algebra Vector algebra and geometric algebra are alternative approaches to providing additional algebraic structures on vector spaces, with geometric interpretations, particularly vector fields in multivariable calculus and applications in mathematical physics. Vector algebra is specific to Euclidean 3-space, while geometric algebra uses multilinear algebra and applies in all dimensions and signatures, notably 3+1 spacetime as well as 2 dimensions. They are mathematically equivalent in 3 dimensions, though the approaches differ. Vector algebra is more widely used in elementary multivariable calculus, while geometric algebra is used in some more advanced treatments, and is proposed for elementary use as well. In advanced mathematics, particularly differential geometry, neither is widely used, with differential forms being far more widely used.

50.1 Basic concepts and operations In vector algebra the basic objects are scalars and vectors, and the operations (beyond the vector space operations of scalar multiplication and vector addition) are the dot (or scalar) product and the cross product ×. In geometric algebra the basic objects are multivectors (scalars are 0-vectors, vectors are 1-vectors, etc.), and the operations include the Clifford product (here called “geometric product”) and the exterior product. The dot product/inner product/scalar product is defined on 1-vectors, and allows the geometric product to be expressed as the sum of the inner product and the exterior product when multiplying 1-vectors. A distinguishing feature is that vector algebra uses the cross product, while geometric algebra uses the exterior product (and the geometric product). More subtly, geometric algebra in Euclidean 3-space distinguishes 0-vectors, 1-vectors, 2-vectors, and 3-vectors, while elementary vector algebra identifies 1-vectors and 2-vectors (as vectors) and 0-vectors and 3-vectors (as scalars), though more advanced vector algebra distinguishes these as scalars, vectors, pseudovectors, and pseudoscalars. Unlike vector algebra, geometric algebra includes sums of k-vectors of differing k. The cross product does not generalize to dimensions other than 3 (as a product of two vectors, yielding a third vector), and in higher dimensions not all k-vectors can be identified with vectors or scalars. By contrast, the exterior product (and geometric product) is defined uniformly for all dimensions and signatures, and multivectors are closed under these operations.

50.2 Embellishments, ad hoc techniques, and tricks More advanced treatments of vector algebra add embellishments to the initial picture – pseudoscalars and pseudovectors (in terms of geometric algebra in 3 dimensions, correspondingly 3-vectors and 2-vectors), while applications to other dimensions use ad hoc techniques and “tricks” rather than a general mathematical approach. By contrast, geometric algebra begins with a complete picture, and applies uniformly in all dimensions. For example, applying vector calculus in 2 dimensions, such as to compute torque or curl, requires adding an artificial 146

50.3. LIST OF ANALOGOUS FORMULAS

147

3rd dimension and extending the vector field to be constant in that dimension. The torque or curl is then a normal vector field in this 3rd dimension. By contrast, geometric algebra in 2 dimensions defines torque and curl as pseudoscalar fields (2-vector fields), without requiring a 3rd dimension. Similarly, the scalar triple product is ad hoc, and can instead be expressed uniformly using the exterior product and the geometric product.

50.3 List of analogous formulas Here are some comparisons between standard R3 vector relations and their corresponding wedge product and geometric product equivalents. All the wedge and geometric product equivalents here are good for more than three dimensions, and some also for two. In two dimensions the cross product is undefined even if what it describes (like torque) is perfectly well defined in a plane without introducing an arbitrary normal vector outside of the space. Many of these relationships only require the introduction of the wedge product to generalize, but since that may not be familiar to somebody with only a traditional background in vector algebra and calculus, some examples are given.

50.3.1

Algebraic and geometric properties of cross and wedge products

Cross and wedge products are both antisymmetric:

v × u = −(u × v) v ∧ u = −(u ∧ v) They are both linear in the first operand

(u + v) × w = u × w + v × w (u + v) ∧ w = u ∧ w + v ∧ w and in the second operand

u × (v + w) = u × v + u × w u ∧ (v + w) = u ∧ v + u ∧ w In general, the cross product is not associative, while the wedge product is

(u × v) × w ̸= u × (v × w) (u ∧ v) ∧ w = u ∧ (v ∧ w) Both the cross and wedge products of two identical vectors are zero:

u×u=0 u∧u=0 u × v is perpendicular to the plane containing u and v . u ∧ v is an oriented representation of the same plane. The cross product of traditional vector algebra (on R3 ) finds its place in geometric algebra G3 as a scaled exterior product

a × b = −i(a ∧ b)

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(this is antisymmetric). Relevant is the distinction between axial and polar vectors in vector algebra, which is natural in geometric algebra as the distinction between vectors and bivectors (elements of grade two). The i here is a unit pseudoscalar of Euclidean 3-space, which establishes a duality between the vectors and the bivectors, and is named so because of the expected property

i2 = (e1 e2 e3 )2 = e1 e2 e3 e1 e2 e3 = −e1 e2 e1 e3 e2 e3 = e1 e1 e2 e3 e2 e3 = −e3 e2 e2 e3 = −1 The equivalence of the R3 cross product and the wedge product expression above can be confirmed by direct multiplication of −i = −e1 e2 e3 with a determinant expansion of the wedge product ∑

u∧v =

∑

(ui vj − vi uj )ei ∧ ej =

1≤i 1, the set of all real numbers of the form a + b√n with a and b integers is a subring of R and hence an integral domain. • For each integer n > 0 the set of all complex numbers of the form a + bi√n with a and b integers is a subring of C and hence an integral domain. In the case n = 1 this integral domain is called the Gaussian integers. • The ring of p-adic integers is an integral domain. • If U is a connected open subset of the complex plane C, then the ring H(U) consisting of all holomorphic functions f : U → C is an integral domain. The same is true for rings of analytic functions on connected open subsets of analytic manifolds. • A regular local ring is an integral domain. In fact, a regular local ring is a UFD.[7][8]

121.3 Non-examples The following rings are not integral domains. • The ring of n × n matrices over any nonzero ring when n ≥ 2. • The ring of continuous functions on the unit interval. • The quotient ring Z/mZ when m is a composite number. • The product ring Z × Z. • The zero ring in which 0=1. • The tensor product C ⊗R C (since, for example, (i ⊗ 1 − 1 ⊗ i) (i ⊗ 1 + 1 ⊗ i) = 0 ). • The quotient ring k[x, y]/(xy) for any field k , since (xy) is not a prime ideal.

121.4 Divisibility, prime elements, and irreducible elements See also: Divisibility (ring theory) In this section, R is an integral domain. Given elements a and b of R, we say that a divides b, or that a is a divisor of b, or that b is a multiple of a, if there exists an element x in R such that ax = b. The elements that divide 1 are called the units of R; these are precisely the invertible elements in R. Units divide all other elements. If a divides b and b divides a, then we say a and b are associated elements or associates.[9] Equivalently, a and b are associates if a=ub for some unit u. If q is a nonzero non-unit, we say that q is an irreducible element if q cannot be written as a product of two non-units.

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If p is a nonzero non-unit, we say that p is a prime element if, whenever p divides a product ab, then p divides a or p divides b. Equivalently, an element p is prime if and only if the principal ideal (p) is a nonzero prime ideal. The notion of prime element generalizes the ordinary definition of prime number in the ring Z, except that it allows for negative prime elements. Every prime [√ ]element is irreducible. The converse is not true in general: for example, in the quadratic integer ring Z −5 the element 3 is irreducible (if it factored nontrivially, the factors would each have to have norm 2 2 (3, but√there) (are no√norm ) 3 elements since a + 5b = 3 has no integer solutions), but not prime (since 3 divides 2 + −5 2 − −5 without dividing either factor). In a unique factorization domain (or more generally, a GCD domain), an irreducible element is a prime element. [√ ] While unique factorization does not hold in Z −5 , there is unique factorization of ideals. See Lasker–Noether theorem.

121.5 Properties • A commutative ring R is an integral domain if and only if the ideal (0) of R is a prime ideal. • If R is a commutative ring and P is an ideal in R, then the quotient ring R/P is an integral domain if and only if P is a prime ideal. • Let R be an integral domain. Then there is an integral domain S such that R ⊂ S and S has an element which is transcendental over R. • The cancellation property holds in any integral domain: for any a, b, and c in an integral domain, if a ≠ 0 and ab = ac then b = c. Another way to state this is that the function x ↦ ax is injective for any nonzero a in the domain. • The cancellation property holds for ideals in any integral domain: if xI = xJ, then either x is zero or I = J. • An integral domain is equal to the intersection of its localizations at maximal ideals. • An inductive limit of integral domains is an integral domain.

121.6 Field of fractions Main article: Field of fractions The field of fractions K of an integral domain R is the set of fractions a/b with a and b in R and b ≠ 0 modulo an appropriate equivalence relation, equipped with the usual addition and multiplication operations. It is “the smallest field containing R" in the sense that there is an injective ring homomorphism R → K such that any injective ring homomorphism from R to a field factors through K. The field of fractions of the ring of integers Z is the field of rational numbers Q. The field of fractions of a field is isomorphic to the field itself.

121.7 Algebraic geometry Integral domains are characterized by the condition that they are reduced (that is x2 = 0 implies x = 0) and irreducible (that is there is only one minimal prime ideal). The former condition ensures that the nilradical of the ring is zero, so that the intersection of all the ring’s minimal primes is zero. The latter condition is that the ring have only one minimal prime. It follows that the unique minimal prime ideal of a reduced and irreducible ring is the zero ideal, so such rings are integral domains. The converse is clear: an integral domain has no nonzero nilpotent elements, and the zero ideal is the unique minimal prime ideal. This translates, in algebraic geometry, into the fact that the coordinate ring of an affine algebraic set is an integral domain if and only if the algebraic set is an algebraic variety. More generally, a commutative ring is an integral domain if and only if its spectrum is an integral affine scheme.

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397

121.8 Characteristic and homomorphisms The characteristic of an integral domain is either 0 or a prime number. If R is an integral domain of prime characteristic p, then the Frobenius endomorphism f(x) = x p is injective.

121.9 See also • Integral domains – wikibook link • Dedekind–Hasse norm – the extra structure needed for an integral domain to be principal • Zero-product property

121.10 Notes [1] Bourbaki, p. 116. [2] Dummit and Foote, p. 228. [3] B.L. van der Waerden, Algebra Erster Teil, p. 36, Springer-Verlag, Berlin, Heidelberg 1966. [4] I.N. Herstein, Topics in Algebra, p. 88-90, Blaisdell Publishing Company, London 1964. [5] J.C. McConnel and J.C. Robson “Noncommutative Noetherian Rings” (Graduate studies in Mathematics Vol. 30, AMS) [6] Pages 91–92 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-20155540-0, Zbl 0848.13001 [7] Auslander, Maurice; Buchsbaum, D. A. (1959). “Unique factorization in regular local rings”. Proc. Natl. Acad. Sci. USA 45 (5): 733–734. doi:10.1073/pnas.45.5.733. PMC 222624. PMID 16590434. [8] Masayoshi Nagata (1958). “A general theory of algebraic geometry over Dedekind domains. II”. Amer. J. Math. (The Johns Hopkins University Press) 80 (2): 382–420. doi:10.2307/2372791. JSTOR 2372791. [9] Durbin, John R. (1993). Modern Algebra: An Introduction (3rd ed.). John Wiley and Sons. p. 224. ISBN 0-471-51001-7. Elements a and b of [an integral domain] are called associates if a | b and b | a.

121.11 References • Adamson, Iain T. (1972). Elementary rings and modules. University Mathematical Texts. Oliver and Boyd. ISBN 0-05-002192-3. • Bourbaki, Nicolas (1998). Algebra, Chapters 1–3. Berlin, New York: Springer-Verlag. ISBN 978-3-54064243-5. • Mac Lane, Saunders; Birkhoff, Garrett (1967). Algebra. New York: The Macmillan Co. ISBN 1-56881-0687. MR 0214415. • Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). New York: Wiley. ISBN 978-0471-43334-7. • Hungerford, Thomas W. (1974). Algebra. New York: Holt, Rinehart and Winston, Inc. ISBN 0-03-030558-6. • Lang, Serge (2002). Algebra. Graduate Texts in Mathematics 211. Berlin, New York: Springer-Verlag. ISBN 978-0-387-95385-4. MR 1878556. • Sharpe, David (1987). Rings and factorization. Cambridge University Press. ISBN 0-521-33718-6. • Rowen, Louis Halle (1994). Algebra: groups, rings, and fields. A K Peters. ISBN 1-56881-028-8. • Lanski, Charles (2005). Concepts in abstract algebra. AMS Bookstore. ISBN 0-534-42323-X.

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• Milies, César Polcino; Sehgal, Sudarshan K. (2002). An introduction to group rings. Springer. ISBN 1-40200238-6. • B.L. van der Waerden, Algebra, Springer-Verlag, Berlin Heidelberg, 1966.

Chapter 122

Integral element In commutative algebra, an element b of a commutative ring B is said to be integral over A, a subring of B, if there are n ≥ 1 and aj ∈ A such that bn + an−1 bn−1 + · · · + a1 b + a0 = 0. That is to say, b is a root of a monic polynomial over A.[1] If every element of B is integral over A, then it is said that B is integral over A, or equivalently B is an integral extension of A. If A, B are fields, then the notions of “integral over” and of an “integral extension” are precisely “algebraic over” and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial). The special case of an integral element of greatest interest in number √ theory is that of complex numbers integral over Z; in this context, they are usually called algebraic integers (e.g., 2 ). The algebraic integers in a finite extension field k of the rationals Q form a subring of k, called the ring of integers of k, a central object of study in algebraic number theory. The set of elements of B that are integral over A is called the integral closure of A in B. It is a subring of B containing A. In this article, the term ring will be understood to mean commutative ring with a unity, occasionally given the moniker community.

122.1 Examples • Integers are the only elements of Q that are integral over Z. In other words, Z is the integral closure of Z in Q. √ √ • Gaussian integers, complex numbers of the form a + b −1, a, b ∈ Z , are integral over Z. Z[ −1] is then √ the integral closure of Z in Q( −1) . √ √ • The integral closure of Z in Q( 5) consists of elements of form (a + b 5)/2 , where a and b are integers and a2 − 5b2 is multiple of 4; this example and the previous one are examples of quadratic integers. • Let ζ be a root of unity. Then the integral closure of Z in the cyclotomic field Q(ζ) is Z[ζ].[2] • The integral closure of Z in the field of complex numbers C is called the ring of algebraic integers. • If k is an algebraic closure of a field k, then k[x1 , . . . , xn ] is integral over k[x1 , . . . , xn ]. • Let a finite group G act on a ring A. Then A is integral over AG the set of elements fixed by G. see ring of invariants. • The roots of unity, nilpotent elements and idempotent elements in any ring are integral over Z. • Let R be a ring and u a unit in a ring containing R. Then[3] 1. u−1 is integral over R if and only if u−1 ∈ R[u]. 399

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2. R[u] ∩ R[u−1 ] is integral over R. • The integral closure of C[[x]] in a finite extension of C((x)) is of the form C[[x1/n ]] (cf. Puiseux series) • The integral closure of the homogeneous coordinate ring of a normal projective variety X is the ring of sections[4] ⊕ n≥0

H0 (X, OX (n)).

122.2 Equivalent definitions Let B be a ring, and let A be a subring of B. Given an element b in B, the following conditions are equivalent: (i) b is integral over A; (ii) the subring A[b] of B generated by A and b is a finitely generated A-module; (iii) there exists a subring C of B containing A[b] and which is a finitely-generated A-module; (iv) there exists a finitely generated A-submodule M of B such that bM ⊂ M and M is not annihilated by any nonzero polynomial in A[b]. The usual proof of this uses the following variant of the Cayley–Hamilton theorem on determinants: Theorem Let u be an endomorphism of an A-module M generated by n elements and I an ideal of A such that u(M ) ⊂ IM . Then there is a relation: un + a1 un−1 + · · · + an−1 u + an = 0, ai ∈ I i . This theorem (with I = A and u multiplication by b) gives (iv) ⇒ (i) and the rest is easy. Coincidentally, Nakayama’s lemma is also an immediate consequence of this theorem. It follows from the above that the set of elements of B that are integral over A forms a subring of B containing A.[5] It is called the integral closure of A in B.[6] If A happens to be the integral closure of A in B, then A is said to be integrally closed in B. If B is the total ring of fractions of A (e.g., the field of fractions when A is an integral domain), then one sometimes drops qualification “in B” and simply says “integral closure” and "integrally closed.”[7] Let A be an integral domain with the field of fractions K and A' the integral closure of A in an algebraic field extension L of K. Then the field of fractions of A' is L. In particular, A' is an integrally closed domain. Similarly, “integrality” is transitive. Let C be a ring containing B and c in C. If c is integral over B and B integral over A, then c is integral over A. In particular, if C is itself integral over B and B is integral over A, then C is also integral over A. Note that (iii) implies that if B is integral over A, then B is a union (equivalently an inductive limit) of subrings that are finitely generated A-modules. If A is noetherian, (iii) can be weakened to: (iii) bis There exists a finitely generated A-submodule of B that contains A[b]. Finally, the assumption that A be a subring of B can be modified a bit. If f: A → B is a ring homomorphism, then one says f is integral if B is integral over f(A). In the same way one says f is finite (B finitely generated A-module) or of finite type (B finitely generated A-algebra). In this viewpoint, one says that f is finite if and only if f is integral and of finite-type. Or more explicitly, B is a finitely generated A-module if and only if B is generated as A-algebra by a finite number of elements integral over A.

122.3. INTEGRAL EXTENSIONS

401

122.3 Integral extensions An integral extension A⊆B has the going-up property, the lying over property, and the incomparability property (Cohen-Seidenberg theorems). Explicitly, given a chain of prime ideals p1 ⊂ · · · ⊂ pn in A there exists a p′1 ⊂ · · · ⊂ p′n in B with pi = p′i ∩ A (going-up and lying over) and two distinct prime ideals with inclusion relation cannot contract to the same prime ideal (incomparability). In particular, the Krull dimensions of A and B are the same. Furthermore, if A is an integrally closed domain, then the going-down holds (see below). In general, the going-up implies the lying-over.[8] Thus, in the below, we simply say the “going-up” to mean “going-up” and “lying-over”. When A, B are domains such that B is integral over A, A is a field if and only if B is a field. As a corollary, one has: given a prime ideal q of B, q is a maximal ideal of B if and only if q ∩ A is a maximal ideal of A. Another corollary: if L/K is an algebraic extension, then any subring of L containing K is a field. Let B be a ring that is integral over a subring A and k an algebraically closed field. If f : A → k is a homomorphism, then f extends to a homomorphism B → k.[9] This follows from the going-up. Let f : A → B be an integral extension of rings. Then the induced map

f # : Spec B → Spec A,

p 7→ f −1 (p)

is a closed map; in fact, f # (V (I)) = V (f −1 (I)) for any ideal I and f # is surjective if f is injective. This is a geometric interpretation of the going-up. If B is integral over A, then B ⊗A R is integral over R for any A-algebra R.[10] In particular, Spec(B ⊗A R) → Spec R is closed; i.e., the integral extension induces a “universally closed” map. This leads to a geometric characterization of integral extension. Namely, let B be a ring with only finitely many minimal prime ideals (e.g., integral domain or noetherian ring). Then B is integral over a (subring) A if and only if Spec(B ⊗A R) → Spec R is closed for any A-algebra R.[11] Let A be an integrally closed domain with the field of fractions K, L a finite normal extension of K, B the integral closure of A in L. Then the group G = Gal(L/K) acts transitively on each fiber of Spec B → Spec A . (Proof: Suppose p2 ̸= σ(p1 ) for any σ in ∏G. Then, by prime avoidance, there is an element x in p2 such that σ(x) ̸∈ p1 for any σ . G fixes the element y = σ σ(x) and thus y is purely inseparable over K. Then some power y e belongs to K; in fact, to A since A is integrally closed. Thus, we found y e is in p2 ∩ A but not in p1 ∩ A ; i.e., p1 ∩ A ̸= p2 ∩ A .) Remark: The same idea in the proof shows that if L/K is a purely inseparable extension (need not be normal), then Spec B → Spec A is bijective. Let A, K, etc. as before but assume L is only a finite field extension of K. Then (i) Spec B → Spec A has finite fibers. (ii) the going-down holds between A and B: given p1 ⊂ · · · ⊂ pn−1 ⊂ pn = p′n ∩ A , there exists p′1 ⊂ · · · ⊂ p′n that contracts to it. Indeed, in both statements, by enlarging L, we can assume L is a normal extension. Then (i) is immediate. As for (ii), by the going-up, we can find a chain p′′i that contracts to p′i . By transitivity, there is σ ∈ G such that σ(p′′n ) = p′n and then p′i = σ(p′′i ) are the desired chain. Let B be a ring and A a subring that is a noetherian integrally closed domain (i.e., Spec A is a normal scheme.) If B is integral over A, then Spec B → Spec A is submersive; i.e., the topology of Spec A is the quotient topology.[12] The proof uses the notion of constructible sets. (See also: torsor (algebraic geometry).)

122.4 Integral closure See also: Integral closure of an ideal Let A ⊂ B be rings and A' the integral closure of A in B. (See above for the definition.)

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Integral closures behave nicely under various constructions. Specifically, for a multiplicatively closed subset S of A, the localization S −1 A' is the integral closure of S −1 A in S −1 B, and A′ [t] ∏ is the integral A[t] in B[t] .[13] ∏ closure ∏ of ′ If Ai are subrings of rings Bi , 1 ≤ i ≤ n , then the integral closure of Ai in Bi is Ai where Ai ′ are the integral closures of Ai in Bi .[14] The integral closure of a local ring A in, say, B, need not be local. (If this is the case, the ring is called unibranch.) This is the case for example when A is Henselian and B is a field extension of the field of fractions of A. If A is a subring of a field K, then the integral closure of A in K is the intersection of all valuation rings of K containing A. Let B be an N -graded subring of an N -graded ring A. Then the integral closure of A in B is an N -graded subring of B.[15] There is also a concept of the integral closure of an ideal. The integral closure of an ideal I ⊂ R , usually denoted by I , is the set of all elements r ∈ R such that there exists a monic polynomial xn + a1 xn−1 + · · · + an−1 x1 + an with ai ∈ I i with r as a root. Note this is the definition that appears, for example, in Eisenbud and is different from Bourbaki’s and Atiyah–MacDonald’s definition. For noetherian rings, there are alternate definitions as well. • r ∈ I if there exists a c ∈ R not contained in any minimal prime, such that crn ∈ I n for all n ≥ 1 . • r ∈ I if in the normalized blow-up of I, the pull back of r is contained in the inverse image of I. The blow-up of an ideal is an operation of schemes which replaces the given ideal with a principal ideal. The normalization of a scheme is simply the scheme corresponding to the integral closure of all of its rings. The notion of integral closure of an ideal is used in some proofs of the going-down theorem.

122.5 Conductor Let B be a ring and A a subring of B such that B is integral over A. Then the annihilator of the A-module B/A is called the conductor of A in B. Because the notion has origin in algebraic number theory, the conductor is denoted by f = f(B/A) . Explicitly, f consists of elements a in A such that aB ⊂ A . (cf. idealizer in abstract algebra.) It is the largest ideal of A that is also an ideal of B.[16] If S is a multiplicatively closed subset of A, then S −1 f(B/A) = f(S −1 B/S −1 A) If B is a subring of the total ring of fractions of A, then we may identify

f(B/A) = HomA (B, A) Example: Let k be a field and let A = k[t2 , t3 ] ⊂ B = k[t] (i.e., A is the coordinate ring of the affine curve x2 = y 3 .) B is the integral closure of A in k(t) . The conductor of A in B is the ideal (t2 , t3 )A . More generally, the conductor of A = k[[ta , tb ]] , a, b relatively prime, is (tc , tc+1 , . . . )A with c = (a − 1)(b − 1) .[17] Suppose B is the integral closure of an integral domain A in the field of fractions of A such that the A-module B/A is finitely generated. Then the conductor f of A is an ideal defining the support of B/A ; thus, A coincides with B in the complement of V (f) in Spec A . In particular, the set {p ∈ Spec A|Ap closed integrally is } , the complement of V (f) , is an open set.

122.6 Finiteness of integral closure An important but difficult question is on the finiteness of the integral closure of a finitely generated algebra. There are several known results.

122.7. NOETHER’S NORMALIZATION LEMMA

403

The integral closure of a Dedekind domain in a finite extension of the field of fractions is a Dedekind domain; in particular, a noetherian ring. This is a consequence of the Krull–Akizuki theorem. In general, the integral closure of a noetherian domain of dimension at most 2 is noetherian; Nagata gave an example of dimension 3 noetherian domain whose integral closure is not noetherian.[18] A nicer statement is this: the integral closure of a noetherian domain is a Krull domain (Mori–Nagata theorem). Nagata also gave an example of dimension 1 noetherian local domain such that the integral closure is not finite over that domain. Let A be a noetherian integrally closed domain with field of fractions K. If L/K is a finite separable extension, then the integral closure A′ of A in L is a finitely generated A-module.[19] This is easy and standard (uses the fact that the trace defines a non-degenerate bilinear form.) Let A be a finitely generated algebra over a field k that is an integral domain with field of fractions K. If L is a finite extension of K, then the integral closure A′ of A in L is a finitely generated A-module and is also a finitely generated k-algebra.[20] The result is due to Noether and can be shown using the Noether normalization lemma as follows. It is clear that it is enough to show the assertion when L/K is either separable or purely inseparable. The separable case is noted above; thus, assume L/K is purely inseparable. By the normalization lemma, A is integral over the polynomial ring S = k[x1 , ..., xd ] . Since L/K is a finite purely inseparable extension, there is a power q of a prime number such that every element of L is a q-th root of an element in K. Let k ′ be a finite extension of k containing all q-th roots of 1/q 1/q coefficients of finitely many rational functions that generate L. Then we have: L ⊂ k ′ (x1 , ..., xd ). The ring on 1/q 1/q the right is the field of fractions of k ′ [x1 , ..., xd ] , which is the integral closure of S; thus, contains A′ . Hence, ′ A is finite over S; a fortiori, over A. The result remains true if we replace k by Z. The integral closure of a complete local noetherian domain A in a finite extension of the field of fractions of A is finite over A.[21] More precisely, for a local noetherian ring R, we have the following chains of implications:[22] (i) A complete ⇒ A is a Nagata ring b is (ii) A is a Nagata domain ⇒ A analytically unramified ⇒ the integral closure of the completion A b finite over A ⇒ the integral closure of A is finite over A.

122.7 Noether’s normalization lemma Main article: Noether normalization lemma Noether’s normalisation lemma is a theorem in commutative algebra. Given a field K and a finitely generated Kalgebra A, the theorem says it is possible to find elements y1 , y2 , ..., ym in A that are algebraically independent over K such that A is finite (and hence integral) over B = K[y1 ,..., ym]. Thus the extension K ⊂ A can be written as a composite K ⊂ B ⊂ A where K ⊂ B is a purely transcendental extension and B ⊂ A is finite.[23]

122.8 Notes [1] The above equation is sometimes called an integral equation and b is said to be integrally dependent on A (as opposed to algebraic dependent.) [2] Milne ANT, Theorem 6.4 [3] Kaplansky, 1.2. Exercise 4. [4] Hartshorne 1977, Ch. II, Exercise 5.14 [5] Proof: If x, y are elements of B that are integral over A, then x+y, xy, −x are integral over A since they stabilize A[x]A[y] , which is a finitely generated module over A and is annihilated only by zero. [6] The proof is due to Dedekind (Milne, ANT). Alternatively, one can use symmetric polynomials to show integral elements form a ring. (loc cit.) [7] Chapter 2 of Huneke and Swanson 2006 [8] Kaplansky 1970, Theorem 42 [9] Bourbaki 2006, Ch 5, §2, Corollary 4 to Theorem 1.

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[10] Bourbaki 2006, Ch 5, §1, Proposition 5 [11] Atiyah-MacDonald 1969, Ch 5. Exercise 35 [12] Matsumura 1970, Ch 2. Theorem 7 [13] An exercise in Atiyah–MacDonald. [14] Bourbaki 2006, Ch 5, §1, Proposition 9 [15] Proof: Let ϕ : B → B[t] be a ring homomorphism such that ϕ(bn ) = bn tn if bn is homogeneous of degree n. The integral closure of A[t] in B[t] is A′ [t] , where A′ is the integral closure of A in B. If b in B is integral over A, then ϕ(b) is integral over A[t] ; i.e., it is in A′ [t] . That is, each coefficient bn in the polynomial ϕ(b) is in A'. [16] Chapter 12 of Huneke and Swanson 2006 [17] Swanson 2006, Example 12.2.1 [18] Swanson 2006, Exercise 4.9 [19] Atiyah-MacDonald 1969, Ch 5. Proposition 5.17 [20] Hartshorne 1977, Ch I. Theorem 3.9 A [21] Swanson 2006, Theorem 4.3.4 [22] Matsumura 1970, Ch 12 [23] Chapter 4 of Reid.

122.9 References • M. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison–Wesley, 1994. ISBN 0-201-407515 • Nicolas Bourbaki, Algèbre commutative, 2006. • Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8. • Kaplansky, Irving (September 1974). Commutative Rings. Lectures in Mathematics. University of Chicago Press. ISBN 0-226-42454-5. • Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics 52, New York: SpringerVerlag, ISBN 978-0-387-90244-9, MR 0463157 • Matsumura, H (1970), Commutative algebra • H. Matsumura Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8. • J. S. Milne, “Algebraic number theory.” available at http://www.jmilne.org/math/ • Huneke, Craig; Swanson, Irena (2006), Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series 336, Cambridge, UK: Cambridge University Press, ISBN 978-0-521-68860-4, MR 2266432 • M. Reid, Undergraduate Commutative Algebra, London Mathematical Society, 29, Cambridge University Press, 1995.

122.10 Further reading • Irena Swanson, Integral closures of ideals and rings

Chapter 123

Invariant basis number In mathematics, more specifically in the field of ring theory, a ring has the invariant basis number (IBN) property if all finitely generated free left modules over R have a well-defined rank. In the case of fields, the IBN property becomes the statement that finite-dimensional vector spaces have a unique dimension.

123.1 Definition A ring R has invariant basis number (IBN) if for all positive integers m and n, Rm isomorphic to Rn (as left Rmodules) implies that m = n. Equivalently, this means there do not exist distinct positive integers m and n such that Rm is isomorphic to Rn . Rephrasing the definition of invariant basis number in terms of matrices, it says that, whenever A is an m-by-n matrix over R and B is an n-by-m matrix over R such that AB = I and BA = I, then m = n. This form reveals that the definition is left-right symmetric, so it makes no difference whether we define IBN in terms of left or right modules; the two definitions are equivalent. Note that the isomorphisms in the definitions are not ring isomorphisms, they are module isomorphisms.

123.2 Discussion The main purpose of the invariant basis number condition is that free modules over an IBN ring satisfy an analogue of the dimension theorem for vector spaces: any two bases for a free module over an IBN ring have the same cardinality. Assuming the ultrafilter lemma (a strictly weaker form of the axiom of choice), this result is actually equivalent to the definition given here, and can be taken as an alternative definition. The rank of a free module Rn over an IBN ring R is defined to be the cardinality of the exponent m of any (and therefore every) R-module Rm isomorphic to Rn . Thus the IBN property asserts that every isomorphism class of free R-modules has a unique rank. The rank is not defined for rings not satisfying IBN. For vector spaces, the rank is also called the dimension. Thus the result above is in short: the rank is uniquely defined for all free R-modules iff it is uniquely defined for finitely generated free R-modules.

123.3 Examples Any field satisfies IBN, and this amounts to the fact that finite-dimensional vector spaces have a well defined dimension. Moreover, any commutative ring (except in the trivial case where 1 = 0) satisfies IBN, as does any left-Noetherian ring and any semilocal ring. Proof Let A be a commutative ring and assume there exists an A-module isomorphism f : An → Ap . Let (e1 , . . . , en ) 405

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the canonical basis of An , which means ei ∈ An is all zeros except a one in i-th position. By Krull’s theorem, let I a maximal proper ideal of A and (i1 , . . . , in ) ∈ I n . An A-module morphism means

f (i1 , . . . , in ) =

n ∑

ik f (ek ) ∈ I p

k=0

( )p ( )n → A , that can easily be proven to be an because I is an ideal. So f induces an A/I-module morphism f ′ : A I I isomorphism. Since A/I is a field, f' is an isomorphism between finite dimensional vector spaces, so n=p. An example of a ring that does not satisfy IBN is the ring of column finite matrices CFMN (R) , the matrices with coefficients in a ring R, with entries indexed by N × N and with each column having only finitely many non-zero entries. That last requirement allows us to define the product of infinite matrices MN, giving the ring structure. A left module isomorphism CFMN (R) ∼ = CFMN (R)2 is given by: ψ : CFMN (R) M

→ 7 →

CFMN (R)2 ( of columns oddM, of columns even M )

This infinite matrix ring turns out to be isomorphic to the endomorphisms of a right free module over R of countable rank, which is found on page 190 of (Hungerford). From this isomorphism, it is possible to show (abbreviating CFMN (R) = S ) that S≅S n for any positive integer n, and hence S n ≅S m for any two positive integers m and n. There are other examples of non-IBN rings without this property, among them Leavitt algebras as seen in (Abrams 2002).

123.4 Other results IBN is a necessary (but not sufficient) condition for a ring with no zero divisors to be embeddable in a division ring (confer field of fractions in the commutative case). See also the Ore condition. Every nontrivial stably finite ring has invariant basis number.

123.5 References Abrams, Gene; Ánh, P. N. (2002), “Some ultramatricial algebras which arise as intersections of Leavitt algebras”, J. Algebra Appl. 1 (4): 357–363, doi:10.1142/S0219498802000227, ISSN 0219-4988, MR 1950131 Hungerford, Thomas W. (1980) [1974], Algebra, Graduate Texts in Mathematics 73, New York: Springer-Verlag, pp. xxiii+502, ISBN 0-387-90518-9, MR 600654 Reprint of the 1974 original

Chapter 124

Invariant factor The invariant factors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain. If R is a PID and M a finitely generated R -module, then M∼ = Rr ⊕ R/(a1 ) ⊕ R/(a2 ) ⊕ · · · ⊕ R/(am ) for some integer r ≥ 0 and a (possibly empty) list of nonzero elements a1 , . . . , am ∈ R for which a1 | a2 | · · · | am . The nonnegative integer r is called the free rank or Betti number of the module M , while a1 , . . . , am are the invariant factors of M and are unique up to associatedness. The invariant factors of a matrix over a PID occur in the Smith normal form and provide a means of computing the structure of a module from a set of generators and relations.

124.1 See also • Elementary divisors

124.2 References • B. Hartley; T.O. Hawkes (1970). Rings, modules and linear algebra. Chapman and Hall. ISBN 0-412-09810-5. Chap.8, p.128. • Chapter III.7, p.153 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-201-55540-0, Zbl 0848.13001

407

Chapter 125

Irreducible element In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.

125.1 Relationship with prime elements Irreducible elements should not be confused with prime elements. (A non-zero non-unit element a in a commutative ring R is called prime if, whenever a|bc for some b and c in R, then a|b or a|c.) In an integral domain, every prime element is irreducible,[1][2] but the converse is not true in general. The converse is true for unique factorization domains[2] (or, more generally, GCD domains.) Moreover, while an ideal generated by a prime element is a prime ideal, it is not true in general that an ideal generated by an irreducible element is an irreducible ideal. However, if D is a GCD domain, and x is an irreducible element of D , then the ideal generated by x is a prime ideal of D .[3]

125.2 Example √ In the quadratic integer ring Z[ −5], it can be shown using norm arguments that the number 3 is irreducible. However, it is not a prime element in this ring since, for example, √ )( √ ) ( 3 | 2 + −5 2 − −5 = 9, but 3 does not divide either of the two factors.[4]

125.3 See also • Irreducible polynomial

125.4 References [1] Consider p a prime that is reducible: p = ab. Then p|ab ⇒ p|a or p|b. Say p|a ⇒ a = pc, then we have p = ab = pcb ⇒ p(1 − cb) = 0. Because R is an integral domain we have cb = 1. So b is a unit and p is irreducible. [2] Sharpe (1987) p.54 [3] http://planetmath.org/encyclopedia/IrreducibleIdeal.html [4] William W. Adams and Larry Joel Goldstein (1976), Introduction to Number Theory, p. 250, Prentice-Hall, Inc., ISBN 0-13-491282-9

408

125.4. REFERENCES

409

• Sharpe, David (1987). Rings and factorization. Cambridge University Press. ISBN 0-521-33718-6. Zbl 0674.13008.

Chapter 126

Irreducible ideal In mathematics, an ideal of a commutative ring is said to be irreducible if it cannot be written as the intersection of two larger ideals.[1] Every prime ideal is irreducible.[2] Every irreducible ideal of a Noetherian ring is a primary ideal,[1] and consequently for Noetherian rings an irreducible decomposition is a primary decomposition. Every primary ideal of a principal ideal domain is an irreducible ideal. Every irreducible ideal is a primal ideal.[3] An element of an integral domain is prime if, and only if, an ideal generated by it is a nonzero prime ideal. This is not true for irreducible ideals: an irreducible ideal may be generated by an element that is not an irreducible element, as is the case in Z for the ideal 4Z : It is not the intersection of two strictly greater ideals. An ideal I of a ring A can be irreducible only if the algebraic set it defines is irreducible (that is, any open subset is dense) for the Zariski topology, or equivalently if the closed space of spec A consisting of prime ideals containing I is irreducible for the spectral topology. The converse is not correct, for example the ideal of polynomials in two variables with vanishing terms of first and second order is not irreducible. If k is an algebraically closed field, choosing the radical of an irreducible ideal of a polynomial ring over k is the same thing as choosing an embedding of the affine variety of its Nullstelle in the affine space.

126.1 See also • irreducible module • irreducible space • Laskerian ring

126.2 References [1] Miyanishi, Masayoshi (1998), Algebraic Geometry, Translations of mathematical monographs 136, American Mathematical Society, p. 13, ISBN 9780821887707. [2] Knapp, Anthony W. (2007), Advanced Algebra, Cornerstones, Springer, p. 446, ISBN 9780817645229. [3] Fuchs, Ladislas (1950), “On primal ideals”, Proceedings of the American Mathematical Society 1: 1–6, doi:10.2307/2032421, MR 0032584. Theorem 1, p. 3.

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Irreducible ring In mathematics, especially in the field of ring theory, the term irreducible ring is used in a few different ways. • A (meet-)irreducible ring is one in which the intersection of two nonzero ideals is always nonzero. • A directly irreducible ring is ring which cannot be written as the direct sum of two nonzero rings. • A subdirectly irreducible ring is a ring with a unique, nonzero minimum two-sided ideal. “Meet-irreducible” rings are referred to as “irreducible rings” in commutative algebra. This article adopts the term “meet-irreducible” in order to distinguish between the several types being discussed. Meet-irreducible rings play an important part in commutative algebra, and directly irreducibe and subdirectly irreducible rings play a role in the general theory of structure for rings. Subdirectly irreducible algebras have also found use in number theory. This article follows the convention that rings have multiplicative identity, but are not necessarily commutative.

127.1 Definitions The terms “meet-reducible”, “directly reducible” and “subdirectly reducible” are used when a ring is not meetirreducible, or not directly irreducible, or not subdirectly irreducible, respectively. The following conditions are equivalent for a commutative ring R: • R is meet-irreducible; • the zero ideal in R is irreducible, i.e. the intersection of two non-zero ideals of A always is non-zero. The following conditions are equivalent for a commutative ring R: • R possesses exactly one minimal prime ideal (this prime ideal may be the zero ideal); • the spectrum of R is irreducible. The following conditions are equivalent for a ring R: • R is directly irreducible; • R has no central idempotents except for 0 and 1. The following conditions are equivalent for a ring R: • R is subdirectly irreducible; 411

412

CHAPTER 127. IRREDUCIBLE RING

• when R is written as a subdirect product of rings, then one of the projections of R onto a ring in the subdirect product is an isomorphism; • The intersection of all nonzero ideals of R is nonzero.

127.2 Examples and properties If R is subdirectly irreducible or meet-irreducible, then it is also directly irreducible, but the converses are not true. • All integral domains are meet-irreducible and subdirectly irreducible. In fact, a commutative ring is a domain if and only if it is both meet-irreducible and reduced. • The quotient ring Z/(4Z) is a ring which has all three senses of irreducibility, but it is not a domain. Its only proper ideal is (2Z)/(4Z), which is maximal, hence prime. The ideal is also minimal. • The direct product of two nonzero rings is never directly irreducible, and hence is never meet-irreducible or subdirectly irreducible. For example, in Z × Z the intersection of the non-zero ideals {0} × Z and Z × {0} is equal to the zero ideal {0} × {0}. • Commutative directly irreducible rings are connected rings; that is, their only idempotent elements are 0 and 1.

127.3 Generalizations Commutative meet-irreducible rings play an elementary role in algebraic geometry, where this concept is generalized to the concept of an irreducible scheme.

Chapter 128

Isotypical representation In group theory, an isotypical or primary representation of a group G is a unitary representation π : G −→ B(H) such that any two subrepresentations have equivalent subsubrepresentations. This is related to the notion of a primary or factor representation of a C*-algebra, or to the factor for a von Neumann algebra: the representation π of G is ′′ isotypical iff π(G) is a factor. This term more generally used in the context of semisimple modules.

128.1 Property One of the interesting property of this notion lies in the fact that two isotypical representations are either quasiequivalent either disjoint (in analogy with the fact that irreducible representations are either unitarily equivalent, either disjoint). This can be understood through the correspondence between factor representations and minimal central projection (in a von Neumann algebra),.[1] Two minimal central projections are then either equal, either orthogonal.

128.2 Example Let G be a compact group. A corollary of the Peter–Weyl theorem has that any unitary representation π : G −→ B(H) on a separable Hilbert space H is a possibly infinite direct sum of finite dimensional irreducible representations. An isotypical representation is a direct sum of the equivalent irreducible representations that appear, possibly multiple times, in H .

128.3 References [1] Dixmier C*-algebras Prop. 5.2.7 p.117 1977

• Mackey • “C* algebras”, Jacques Dixmier, Chapter 5 • “Lie Groups”, Claudio Procesi, def. p. 156. • “Group and symmetries”, Yvette Kosmann-Schwarzbach

413

Chapter 129

Jacobian ideal In mathematics the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ. Let O(x1 , . . . , xn ) denote the ring of smooth functions and f a function in the ring. The Jacobian ideal of f is ⟨ Jf :=

∂f ∂f ,..., ∂x1 ∂xn

⟩ .

129.1 See also • Milnor number • Unfolding

414

Chapter 130

Jacobson density theorem In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring R.[1] The theorem can be applied to show that any primitive ring can be viewed as a “dense” subring of the ring of linear transformations of a vector space.[2][3] This theorem first appeared in the literature in 1945, in the famous paper “Structure Theory of Simple Rings Without Finiteness Assumptions” by Nathan Jacobson.[4] This can be viewed as a kind of generalization of the Artin-Wedderburn theorem's conclusion about the structure of simple Artinian rings.

130.1 Motivation and formal statement Let R be a ring and let U be a simple right R-module. If u is a non-zero element of U, u • R = U (where u • R is the cyclic submodule of U generated by u). Therefore, if u, v are non-zero elements of U, there is an element of R that induces an endomorphism of U transforming u to v. The natural question now is whether this can be generalized to arbitrary (finite) tuples of elements. More precisely, find necessary and sufficient conditions on the tuple (x1 , ..., xn) and (y1 , ..., yn) separately, so that there is an element of R with the property that xi • r = yi for all i. If D is the set of all R-module endomorphisms of U, then Schur’s lemma asserts that D is a division ring, and the Jacobson density theorem answers the question on tuples in the affirmative, provided that the xᵢ are linearly independent over D. With the above in mind, the theorem may be stated this way: The Jacobson Density Theorem. Let U be a simple right R-module, D = End(UR), and X ⊂ U a finite and D-linearly independent set. If A is a D-linear transformation on U then there exists r ∈ R such that A(x) = x • r for all x in X.[5]

130.2 Proof In the Jacobson density theorem, the right R-module U is simultaneously viewed as a left D-module where D = End(UR), in the natural way: g • u = g(u). It can be verified that this is indeed a left module structure on U.[6] As noted before, Schur’s lemma proves D is a division ring if U is simple, and so U is a vector space over D. The proof also relies on the following theorem proven in (Isaacs 1993) p. 185: Theorem. Let U be a simple right R-module, D = End(UR), and X ⊂ U a finite set. Write I = annR(X) for the annihilator of X in R. Let u be in U with u • I = 0. Then u is in XD; the D-span of X.

130.2.1

Proof of the Jacobson density theorem

We use induction on |X|. If X is empty, then the theorem is vacuously true and the base case for induction is verified. Assume X is non-empty, let x be an element of X and write Y = X \{x}. If A is any D-linear transformation on U, by the induction hypothesis there exists s ∈ R such that A(y) = y • s for all y in Y. Write I = annR(Y). It is easily seen 415

416

CHAPTER 130. JACOBSON DENSITY THEOREM

that x • I is a submodule of U. If x • I = 0, then the previous theorem implies that x would be in the D-span of Y, contradicting the D-linear independence of X, therefore x • I ≠ 0. Since U is simple, we have: x • I = U. Since A(x) − x • s ∈ U = x • I, there exists i in I such that x • i = A(x) − x • s. Define r = s + i and observe that for all y in Y we have: y · r = y · (s + i) =y·s+y·i =y·s

( sincei ∈ annR (Y ))

= A(y) Now we do the same calculation for x: x · r = x · (s + i) =x·s+x·i = x · s + (A(x) − x · s) = A(x) Therefore, A(z) = z • r for all z in X, as desired. This completes the inductive step of the proof. It follows now from mathematical induction that the theorem is true for finite sets X of any size.

130.3 Topological characterization A ring R is said to act densely on a simple right R-module U if it satisfies the conclusion of the Jacobson density theorem.[7] There is a topological reason for describing R as “dense”. Firstly, R can be identified with a subring of End(DU) by identifying each element of R with the D linear transformation it induces by right multiplication. If U is given the discrete topology, and if UU is given the product topology, and End(DU) is viewed as a subspace of UU and is given the subspace topology, then R acts densely on U if and only if R is dense set in End(DU) with this topology.[8]

130.4 Consequences The Jacobson density theorem has various important consequences in the structure theory of rings.[9] Notably, the Artin–Wedderburn theorem's conclusion about the structure of simple right Artinian rings is recovered. The Jacobson density theorem also characterizes right or left primitive rings as dense subrings of the ring of D-linear transformations on some D-vector space U, where D is a division ring.[3]

130.5 Relations to other results This result is related to the Von Neumann bicommutant theorem, which states that, for a *-algebra A of operators on a Hilbert space H, the double commutant A′′ can be approximated by A on any given finite set of vectors. See also the Kaplansky density theorem in the von Neumann algebra setting.

130.6 Notes [1] Isaacs, p. 184 [2] Such rings of linear transformations are also known as full linear rings. [3] Isaacs, Corollary 13.16, p. 187 [4] Jacobson, Nathan “Structure Theory of Simple Rings Without Finiteness Assumptions”

130.7. REFERENCES

417

[5] Isaacs, Theorem 13.14, p. 185 [6] Incidentally it is also a D-R bimodule structure. [7] Herstein, Definition, p. 40 [8] It turns out this topology is the same as the compact-open topology in this case. Herstein, p. 41 uses this description. [9] Herstein, p. 41

130.7 References • I.N. Herstein (1968). Noncommutative rings (1st edition ed.). The Mathematical Association of America. ISBN 0-88385-015-X. • I. Martin Isaacs (1993). Algebra, a graduate course (1st edition ed.). Brooks/Cole Publishing Company. ISBN 0-534-19002-2. • Jacobson, N. (1945), “Structure theory of simple rings without finiteness assumptions”, Trans. Amer. Math. Soc. 57: 228–245, doi:10.1090/s0002-9947-1945-0011680-8, ISSN 0002-9947, MR 0011680

130.8 External links • PlanetMath page

Chapter 131

Jacobson radical In mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules. It happens that substituting “left” in place of “right” in the definition yields the same ideal, and so the notion is left-right symmetric. The Jacobson radical of a ring is frequently denoted by J(R) or rad(R); however to avoid confusion with other radicals of rings, the former notation will be preferred in this article. The Jacobson radical is named after Nathan Jacobson, who was the first to study it for arbitrary rings in (Jacobson 1945). The Jacobson radical of a ring has numerous internal characterizations, including a few definitions that successfully extend the notion to rings without unity. The radical of a module extends the definition of the Jacobson radical to include modules. The Jacobson radical plays a prominent role in many ring and module theoretic results, such as Nakayama’s lemma.

131.1 Intuitive discussion As with other radicals of rings, the Jacobson radical can be thought of as a collection of “bad” elements. In this case the “bad” property is that these elements annihilate all simple left and right modules of the ring. For purposes of comparison, consider the nilradical of a commutative ring, which consists of all elements that are nilpotent. In fact for any ring, the nilpotent elements in the center of the ring are also in the Jacobson radical.[1] So, for commutative rings, the nilradical is contained in the Jacobson radical. The Jacobson radical is very similar to the nilradical in an intuitive sense. A weaker notion of being bad, weaker than being a zero divisor, is being a non-unit (not invertible under multiplication). The Jacobson radical of a ring consists of elements that satisfy a stronger property than being merely a non-unit – in some sense, a member of the Jacobson radical must not “act as a unit” in any module “internal to the ring.” More precisely, a member of the Jacobson radical must project under the canonical homomorphism to the zero of every “right division ring” (each non-zero element of which has a right inverse) internal to the ring in question. Concisely, it must belong to every maximal right ideal of the ring. These notions are of course imprecise, but at least explain why the nilradical of a commutative ring is contained in the ring’s Jacobson radical. In yet a simpler way, we may think of the Jacobson radical of a ring as method to “mod out bad elements” of the ring – that is, members of the Jacobson radical act as 0 in the quotient ring, R/J(R). If N is the nilradical of commutative ring R, then the quotient ring R/N has no nilpotent elements. Similarly for any ring R, the quotient ring has J(R/J(R))={0} and so all of the “bad” elements in the Jacobson radical have been removed by modding out J(R). Elements of the Jacobson radical and nilradical can be therefore seen as generalizations of 0.

131.2 Equivalent characterizations The Jacobson radical of a ring has various internal and external characterizations. The following equivalences appear in many noncommutative algebra texts such as (Anderson 1992, §15), (Isaacs 1993, §13B), and (Lam 2001, Ch 2). The following are equivalent characterizations of the Jacobson radical in rings with unity (characterizations for rings without unity are given immediately afterward): 418

131.2. EQUIVALENT CHARACTERIZATIONS

419

• J(R) equals the intersection of all maximal right ideals of the ring. It is also true that J(R) equals the intersection of all maximal left ideals within the ring.[2] These characterizations are internal to the ring, since one only needs to find the maximal right ideals of the ring. For example, if a ring is local, and has a unique maximal right ideal, then this unique maximal right ideal is an ideal because it is exactly J(R). Maximal ideals are in a sense easier to look for than annihilators of modules. This characterization is deficient, however, because it does not prove useful when working computationally with J(R). The left-right symmetry of these two definitions is remarkable and has various interesting consequences.[3][2] This symmetry stands in contrast to the lack of symmetry in the socles of R, for it may happen that soc(RR) is not equal to soc(RR). If R is a non-commutative ring, J(R) is not necessarily equal to the intersection of all maximal two-sided ideals of R. For instance, if V is a countable direct sum of copies of a field k and R=End(V) (the ring of endomorphisms of V as a k-module), then J(R)=0 because R is known to be von Neumann regular, but there is exactly one maximal double-sided ideal in R consisting of endomorphisms with finite-dimensional image. (Lam 2001, p. 46, Ex. 3.15) • J(R) equals the sum of all superfluous right ideals (or symmetrically, the sum of all superfluous left ideals) of R. Comparing this with the previous definition, the sum of superfluous right ideals equals the intersection of maximal right ideals. This phenomenon is reflected dually for the right socle of R: soc(RR) is both the sum of minimal right ideals and the intersection of essential right ideals. In fact, these two relationships hold for the radicals and socles of modules in general. • As defined in the introduction, J(R) equals the intersection of all annihilators of simple right R-modules, however it is also true that it is the intersection of annihilators of simple left modules. An ideal that is the annihilator of a simple module is known as a primitive ideal, and so a reformulation of this states that the Jacobson radical is the intersection of all primitive ideals. This characterization is useful when studying modules over rings. For instance, if U is right R-module, and V is a maximal submodule of U, U·J(R) is contained in V, where U·J(R) denotes all products of elements of J(R) (the “scalars”) with elements in U, on the right. This follows from the fact that the quotient module, U/V is simple and hence annihilated by J(R). • J(R) is the unique right ideal of R maximal with the property that every element is right quasiregular.[4][1] Alternatively, one could replace “right” with “left” in the previous sentence.[2] This characterization of the Jacobson radical is useful both computationally and in aiding intuition. Furthermore, this characterization is useful in studying modules over a ring. Nakayama’s lemma is perhaps the most well-known instance of this. Although every element of the J(R) is necessarily quasiregular, not every quasiregular element is necessarily a member of J(R).[1] • While not every quasiregular element is in J(R), it can be shown that y is in J(R) if and only if xy is left quasiregular for all x in R. (Lam 2001, p. 50) • J(R) is the set of all such elements x ∈ R that every element of 1 + RxR is a unit: J(R) = { x ∈ R | 1 + RxR ⊂ R× } . For rings without unity it is possible for R=J(R), however the equation that J(R/J(R))={0} still holds. The following are equivalent characterizations of J(R) for rings without unity appear in (Lam 2001, p. 63): • The notion of left quasiregularity can be generalized in the following way. Call an element a in R left generalized quasiregular if there exists c in R such that c+a-ca= 0. Then J(R) consists of every element a for which ra is left generalized quasiregular for all r in R. It can be checked that this definition coincides with the previous quasiregular definition for rings with unity. • For a ring without unity, the definition of a left simple module M is amended by adding the condition that R•M ≠ 0. With this understanding, J(R) may be defined as the intersection of all annihilators of simple left R modules, or just R if there are no simple left R modules. Rings without unity with no simple modules do exist, in which case R=J(R), and the ring is called a radical ring. By using the generalized quasiregular characterization of the radical, it is clear that if one finds a ring with J(R) nonzero, then J(R) is a radical ring when considered as a ring without unity.

420

CHAPTER 131. JACOBSON RADICAL

131.3 Examples • Rings for which J(R) is {0} are called semiprimitive rings, or sometimes “Jacobson semisimple rings”. The Jacobson radical of any field, any von Neumann regular ring and any left or right primitive ring is {0}. The Jacobson radical of the integers is {0}. • The Jacobson radical of the ring Z/12Z (see modular arithmetic) is 6Z/12Z, which is the intersection of the maximal ideals 2Z/12Z and 3Z/12Z. • If K is a field and R is the ring of all upper triangular n-by-n matrices with entries in K, then J(R) consists of all upper triangular matrices with zeros on the main diagonal. • If K is a field and R = K[[X1 , ..., Xn]] is a ring of formal power series, then J(R) consists of those power series whose constant term is zero. More generally: the Jacobson radical of every local ring is the unique maximal ideal of the ring. • Start with a finite, acyclic quiver Γ and a field K and consider the quiver algebra KΓ (as described in the quiver article). The Jacobson radical of this ring is generated by all the paths in Γ of length ≥ 1. • The Jacobson radical of a C*-algebra is {0}. This follows from the Gelfand–Naimark theorem and the fact for a C*-algebra, a topologically irreducible *-representation on a Hilbert space is algebraically irreducible, so that its kernel is a primitive ideal in the purely algebraic sense (see spectrum of a C*-algebra).

131.4 Properties • If R is unital and is not the trivial ring {0}, the Jacobson radical is always distinct from R since rings with unity always have maximal right ideals. However, some important theorems and conjectures in ring theory consider the case when J(R) = R - “If R is a nil ring (that is, each of its elements is nilpotent), is the polynomial ring R[x] equal to its Jacobson radical?" is equivalent to the open Köthe conjecture. (Smoktunowicz 2006, p. 260, §5) • The Jacobson radical of the ring R/J(R) is zero. Rings with zero Jacobson radical are called semiprimitive rings. • A ring is semisimple if and only if it is Artinian and its Jacobson radical is zero. • If f : R → S is a surjective ring homomorphism, then f(J(R)) ⊆ J(S). • If M is a finitely generated left R-module with J(R)M = M, then M = 0 (Nakayama’s lemma). • J(R) contains all central nilpotent elements, but contains no idempotent elements except for 0. • J(R) contains every nil ideal of R. If R is left or right Artinian, then J(R) is a nilpotent ideal. This can actually be made stronger: If {0} = T0 ⊆ T1 ⊆ · · · ⊆ Tk = R is a composition series for the right R-module R (such a series is sure to exist if R is right artinian, and there is a similar left composition series if R is left artinian), k then (J (R)) = 0 . (Proof: Since the factors Tu /Tu−1 are simple right R-modules, right multiplication by any element of J(R) annihilates these factors. In other words, (Tu /Tu−1 ) · J (R) = 0 , whence Tu · J (R) ⊆ Tu−1 . Consequently, induction over i shows that all nonnegative integers i and u (for which the following makes i sense) satisfy Tu ·(J (R)) ⊆ Tu−i . Applying this to u = i = k yields the result.) Note, however, that in general the Jacobson radical need not consist of only the nilpotent elements of the ring. • If R is commutative and finitely generated as an algebra over either a field or Z, then J(R) is equal to the nilradical of R. • The Jacobson radical of a (unital) ring is its largest superfluous right (equivalently, left) ideal.

131.5. SEE ALSO

421

131.5 See also • Nilradical • Radical of a module • Radical of an ideal • Frattini subgroup

131.6 Notes [1] Isaacs, p. 181. [2] Isaacs, p. 182. [3] Isaacs, Problem 12.5, p. 173 [4] Isaacs, Corollary 13.4, p. 180

131.7 References • Anderson, Frank W.; Fuller, Kent R. (1992), Rings and categories of modules, Graduate Texts in Mathematics 13 (2 ed.), New York: Springer-Verlag, pp. x+376, ISBN 0-387-97845-3, MR 1245487 (94i:16001) • Atiyah, M. F.; Macdonald, I. G. (1969), Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., pp. ix+128, MR 0242802 (39 #4129) • N. Bourbaki. Éléments de Mathématique. • Herstein, I. N. (1994) [1968], Noncommutative rings, Carus Mathematical Monographs 15, Washington, DC: Mathematical Association of America, pp. xii+202, ISBN 0-88385-015-X, MR 1449137 (97m:16001) Reprint of the 1968 original; With an afterword by Lance W. Small • Isaacs, I. M. (1993). Algebra, a graduate course (1st edition ed.). Brooks/Cole Publishing Company. ISBN 0-534-19002-2. • Jacobson, Nathan (1945), “The radical and semi-simplicity for arbitrary rings”, American Journal of Mathematics 67: 300–320, doi:10.2307/2371731, ISSN 0002-9327, MR 12271 • Lam, T. Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics 131 (2 ed.), New York: Springer-Verlag, pp. xx+385, ISBN 0-387-95183-0, MR 1838439 (2002c:16001) • Pierce, Richard S. (1982), Associative algebras, Graduate Texts in Mathematics 88, New York: SpringerVerlag, pp. xii+436, ISBN 0-387-90693-2, MR 674652 (84c:16001) Studies in the History of Modern Science, 9 • Smoktunowicz, Agata (2006), “Some results in noncommutative ring theory”, International Congress of Mathematicians, Vol. II (PDF), European Mathematical Society, pp. 259–269, ISBN 978-3-03719-022-7, MR 2275597

Chapter 132

Jacobson ring Not to be confused with Jacobson semisimple ring. In algebra, a Hilbert ring or a Jacobson ring is a ring such that every prime ideal is an intersection of primitive ideals. For commutative rings primitive ideals are the same as maximal ideals so in this case a Jacobson ring is one in which every prime ideal is an intersection of maximal ideals. Jacobson rings were introduced independently by Krull (1951, 1952), who named them after Nathan Jacobson because of their relation to Jacobson radicals, and by Goldman (1951), who named them Hilbert rings after David Hilbert because of their relation to Hilbert’s Nullstellensatz.

132.1 Jacobson rings and the Nullstellensatz Hilbert’s Nullstellensatz of algebraic geometry is a special case of the statement that the polynomial ring in finitely many variables over a field is a Hilbert ring. A general form of Hilbert’s Nullstellensatz states that if R is a Jacobson ring, then so is any finitely generated R-algebra S. Moreover the pullback of any maximal ideal J of S is a maximal ideal I of R, and S/J is a finite extension of the field R/I. In particular a morphism of finite type of Jacobson rings induces a morphism of the maximal spectrums of the rings. This explains why for algebraic varieties over fields it is often sufficient to work with the maximal ideals rather than with all prime ideals, as was done before the introduction of schemes. For more general rings such as local rings, it is no longer true that morphisms of rings induce morphisms of the maximal spectra, and the use of prime ideals rather than maximal ideals gives a cleaner theory.

132.2 Examples • Any field is a Jacobson ring. • Any principal ideal domain or Dedekind domain with Jacobson radical zero is a Jacobson ring. In principal ideal domains and Dedekind domains, the nonzero prime ideals are already maximal, so the only thing to check is if the zero ideal is an intersection of maximal ideals. Asking for the Jacobson radical to be zero guarantees this. In principal ideal domains and Dedekind domains, the Jacobson radical vanishes if and only if there are infinitely many prime ideals. • Any finitely generated algebra over a Jacobson ring is a Jacobson ring. In particular, any finitely generated algebra over a field or the integers, such as the coordinate ring of any affine algebraic set, is a Jacobson ring. • A local ring has exactly one maximal ideal, so it is a Jacobson ring exactly when that maximal ideal is the only prime ideal. Thus any commutative local ring with Krull dimension zero is Jacobson, but if the Krull dimension is 1 or more, the ring cannot be Jacobson. • (Amitsur 1956) showed that any countably generated algebra over an uncountable field is a Jacobson ring. 422

132.3. CHARACTERIZATIONS

423

132.3 Characterizations The following conditions on a commutative ring R are equivalent: • R is a Jacobson ring • Every prime ideal of R is an intersection of maximal ideals. • Every radical ideal is an intersection of maximal ideals. • Every Goldman ideal is maximal. • Every quotient ring of R by a prime ideal has a zero Jacobson radical. • In every quotient ring, the nilradical is equal to the Jacobson radical. • Every finitely generated algebra over R that is a field is finitely generated as an R-module. (Zariski’s lemma) • Every prime ideal P of R such that R/P has an element x with (R/P)[x−1 ] a field is a maximal prime ideal. • The spectrum of R is a Jacobson space, meaning that every closed subset is the closure of the set of closed points in it. • (For Noetherian rings R): R has no prime ideals P such that R/P is a 1-dimensional semi-local ring.

132.4 Properties • A commutative ring R is a Jacobson ring if and only if R[x], the ring of polynomials over R, is a Jacobson ring.[1]

132.5 Notes [1] Kaplansky, Theorem 31

132.6 References • Amitsur, A. S. (1956), “Algebras over infinite fields”, Proceedings of the American Mathematical Society 7: 35–48, doi:10.2307/2033240, ISSN 0002-9939, MR 0075933 • Commutative algebra by D. Eisenbud, ISBN 0-387-94269-6 • Goldman, Oscar (1951), “Hilbert rings and the Hilbert Nullstellensatz”, Mathematische Zeitschrift 54: 136– 140, doi:10.1007/BF01179855, ISSN 0025-5874, MR 0044510 • Grothendieck, Alexandre; Dieudonné, Jean (1966). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Troisième partie”. Publications Mathématiques de l'IHÉS 28: section 10. doi:10.1007/bf02684343. MR 0217086. • Hazewinkel, Michiel, ed. (2001), “Jacobson_ring”, Encyclopedia of Mathematics, Springer, ISBN 978-155608-010-4 • Kaplansky, Irving (1974), Commutative rings (Revised ed.), University of Chicago Press, ISBN 0-226-424545, MR 0345945 • Krull, Wolfgang (1951), “Jacobsonsche Ringe, Hilbertscher Nullstellensatz, Dimensionstheorie”, Mathematische Zeitschrift 54: 354–387, doi:10.1007/BF01238035, ISSN 0025-5874, MR 0047622 • Krull, Wolfgang (1952), “Jacobsonsches Radikal und Hilbertscher Nullstellensatz”, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950 2, Providence, R.I.: American Mathematical Society, pp. 56–64, MR 0045097

Chapter 133

Jacobson’s conjecture In abstract algebra, Jacobson’s conjecture is an open problem in ring theory concerning the intersection of powers of the Jacobson radical of a Noetherian ring. It has only been proven for special types of Noetherian rings, so far. Examples exist to show that the conjecture can fail when the ring is not Noetherian on a side, so it is absolutely necessary for the ring to be two-sided Noetherian. The conjecture is named for the algebraist Nathan Jacobson who posed the first version of the conjecture.

133.1 Statement For a ring R with Jacobson radical J, the nonnegative powers J n are defined by using the product of ideals. Jacobson’s conjecture: In a right-and-left Noetherian ring,

∩ n∈N

J n = {0}.

In other words: “The only element of a Noetherian ring in all powers of J is 0.” The original conjecture posed by Jacobson in 1956[1] asked about noncommutative one-sided Noetherian rings, however Herstein produced a counterexample in 1965[2] and soon after Jategaonkar produced a different example which was a left principal ideal domain.[3] From that point on, the conjecture was reformulated to require two-sided Noetherian rings.

133.2 Partial results Jacobson’s conjecture has been verified for particular types of Noetherian rings: • Commutative Noetherian rings all satisfy Jacobson’s conjecture. This is a consequence of the Krull intersection theorem. • Fully bounded Noetherian rings[4][5] • Noetherian rings with Krull dimension 1[6] • Noetherian rings satisfying the second layer condition[7]

133.3 References [1] Jacobson, Nathan (1956), Structure of rings, American Mathematical Society, Colloquium Publications, vol. 37, 190 Hope Street, Prov., R. I.: American Mathematical Society, p. 200, MR 0081264. As cited by Brown, K. A.; Lenagan, T. H. (1982), “A note on Jacobson’s conjecture for right Noetherian rings”, Glasgow Mathematical Journal 23 (1): 7–8, doi:10.1017/S0017089500004729, MR 641612.

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133.3. REFERENCES

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[2] Herstein 1965. [3] Jategaonkar 1968. [4] Cauchon 1974. [5] Jategaonkar 1974. [6] Lenagan 1977. [7] Jategaonkar 1982.

• Cauchon, Gérard (1974), “Sur l'intersection des puissances du radical d'un T-anneau noethérien”, C. R. Acad. Sci. Paris Sér. A (in French) 279: 91–93, MR 0347894 • Goodearl, K. R.; Warfield, R. B., Jr. (2004), An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts 61 (2 ed.), Cambridge: Cambridge University Press, pp. xxiv+344, ISBN 0-521-54537-4, MR 2080008 • Herstein, I. N. (1965), “A counterexample in Noetherian rings”, Proc. Nat. Acad. Sci. U.S.A. 54: 1036–1037, doi:10.1073/pnas.54.4.1036, ISSN 0027-8424, MR 0188253 • Jategaonkar, Arun Vinayak (1968), “Left principal ideal domains”, J. Algebra 8: 148–155, doi:10.1016/00218693(68)90040-9, ISSN 0021-8693, MR 0218387 • Jategaonkar, Arun Vinayak (1974), “Jacobson’s conjecture and modules over fully bounded Noetherian rings”, J. Algebra 30: 103–121, doi:10.1016/0021-8693(74)90195-1, ISSN 0021-8693, MR 0352170 • Jategaonkar, Arun Vinayak (1982), “Solvable Lie algebras, polycyclic-by-finite groups and bimodule Krull dimension”, Comm. Algebra 10 (1): 19–69, doi:10.1080/00927878208822700, ISSN 0092-7872, MR 674687 • Lenagan, T. H. (1977), “Noetherian rings with Krull dimension one”, J. London Math. Soc. (2) 15 (1): 41–47, ISSN 0024-6107, MR 0442008 • Rowen, Louis H. (1988), Ring theory. Vol. I, Pure and Applied Mathematics 127, Boston, MA: Academic Press Inc., pp. xxiv+538, ISBN 0-12-599841-4, MR 940245

Chapter 134

Jaffard ring In mathematics, a Jaffard ring is a type of ring, more general than a Noetherian ring, for which Krull dimension behaves as expected in polynomial extensions. They are named for Paul Jaffard who first studied them in 1960. Formally, a Jaffard ring is a ring R such that the polynomial ring

dim R[T1 , . . . , Tn ] = n + dim R, where “dim” denotes Krull dimension. A Jaffard ring that is also an integral domain is called a Jaffard domain. The Jaffard property is satisfied by any Noetherian ring R, so examples of non-Jaffardian rings are quite difficult to find. Nonetheless, an example was given in 1953 by Abraham Seidenberg: the subring of

Q[[T ]] consisting of those formal power series whose constant term is rational.

134.1 References • Bouvier, Alain; Kabbaj, Salah (1988). “Examples of Jaffard domains”. J. Pure Appl. Algebra 54 (2-3): 155– 165. doi:10.1016/0022-4049(88)90027-8. Zbl 0656.13011. • Jaffard, Paul (1960). Théorie de la dimension dans les anneaux de polynômes. Mém. Sci. Math. (in French) 146. Zbl 0096.02502. • Seidenberg, Abraham (1953). “A note on the dimension theory of rings”. Pacific J. Math. 3: 505–512. doi:10.2140/pjm.1953.3.505. ISSN 0030-8730. MR 0054571. Zbl 0052.26902.

134.2 External links • Jaffard ring at PlanetMath.org.

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Chapter 135

Kaplansky’s conjecture The mathematician Irving Kaplansky is notable for proposing numerous conjectures in several branches of mathematics, including a list of ten conjectures on Hopf algebras. They are usually known as Kaplansky’s conjectures.

135.1 Kaplansky’s conjectures on groups rings Let K be a field, and G a torsion-free group. Kaplansky’s zero divisor conjecture states that the group ring K[G] does not contain any zero divisors, that is, it is a domain. No counterexamples have been found and the conjecture has been proved for wide classes of groups. Two related conjectures are: • K[G] does not contain any non-trivial units - if ab = 1 in K[G], then a = k.g for some k in K and g in G. • K[G] does not contain any non-trivial idempotents - if a2 = a, then a = 1 or a = 0. The zero-divisor conjecture implies the idempotent conjecture and is implied by the units conjecture. As of 2014 all three are open. Another related conjecture is the Kadison idempotent conjecture, also known as the Kadison– Kaplansky conjecture, which generalises Kaplansky’s idempotent conjecture to the reduced group C*-algebra. The idempotent and zero-divisor conjectures have implications for geometric group theory. If the Farrell-Jones conjecture holds for K[G] then so does the idempotent conjecture.

135.2 Kaplansky’s conjecture on Banach algebras This conjecture states that every algebra homomorphism from the Banach algebra C(X) (continuous complex-valued functions on X, where X is a compact Hausdorff space) into any other Banach algebra, is necessarily continuous. The conjecture is equivalent to the statement that every algebra norm on C(X) is equivalent to the usual uniform norm. (Kaplansky himself had earlier shown that every complete algebra norm on C(X) is equivalent to the uniform norm.) In the mid-1970s, H. Garth Dales and J. Esterle independently proved that, if one furthermore assumes the validity of the continuum hypothesis, there exist compact Hausdorff spaces X and discontinuous homomorphisms from C(X) to some Banach algebra, giving counterexamples to the conjecture. In 1976, R. M. Solovay (building on work of H. Woodin) exhibited a model of ZFC (Zermelo–Fraenkel set theory + axiom of choice) in which Kaplansky’s conjecture is true. Kaplansky’s conjecture is thus an example of a statement undecidable in ZFC.

135.3 References • H. G. Dales, Automatic continuity: a survey. Bull. London Math. Soc. 10 (1978), no. 2, 129–183. • W. Lück, L2 -Invariants: Theory and Applications to Geometry and K-Theory. Berlin:Springer 2002 ISBN 3-540-43566-2 427

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• D.S. Passman, The Algebraic Structure of Group Rings, Pure and Applied Mathematics, Wiley-Interscience, New York, 1977. ISBN 0-471-02272-1 • M. Puschnigg, The Kadison-Kaplansky conjecture for word-hyperbolic groups. Invent. Math. 149 (2002), no. 1, 153–194. • H. G. Dales and W. H. Woodin, An introduction to independence for analysts, Cambridge 1987

Chapter 136

Kasch ring In ring theory, a subfield of abstract algebra, a right Kasch ring is a ring R for which every simple right R module is isomorphic to a right ideal of R.[1] Analogously the notion of a left Kasch ring is defined, and the two notions are independent of each other. Kasch rings are named in honor of mathematician Friedrich Kasch. Kasch originally called Artinian rings whose proper ideals have nonzero annihilators S-rings. (Kasch 1954)(Morita 1966) The characterizations below show that Kasch rings generalize S-rings.

136.1 Definition Equivalent definitions will be introduced only for the right-hand version, with the understanding that the left-hand analogues are also true. The Kasch conditions have a few equivalences using the concept of annihilators, and this article uses the same notation appearing in the annihilator article. In addition to the definition given in the introduction, the following properties are equivalent definitions for a ring R to be right Kasch. They appear in (Lam 1999, p. 281): 1. For every simple right R module S, there is a nonzero module homomorphism from M into R. 2. The maximal right ideals of R are right annihilators of ring elements, that is, each one is of the form r.ann(x) where x is in R. 3. For any maximal right ideal T of R, ℓ.ann(T ) ̸= {0} . 4. For any proper right ideal T of R, ℓ.ann(T ) ̸= {0} . 5. For any maximal right ideal T of R, r.ann(ℓ.ann(T )) = T . 6. R has no dense right ideals except R itself.

136.2 Examples The content below can be found in references such as (Faith 1999, p. 109), (Lam 1999, §§8C,19B), (Nicholson & Yousif 2003, p.51). • Let R be a semiprimary ring with Jacobson radical J. If R is commutative, or if R/J is a simple ring, then R is right (and left) Kasch. In particular, commutative Artinian rings are right and left Kasch. • For a division ring k, consider a certain subring R of the four-by-four matrix ring with entries from k. The subring R consists of matrices of the following form: 429

430

CHAPTER 136. KASCH RING

 a 0  0 0

0 a 0 0

 b c 0 d  a 0 0 e

This is a right and left Artinian ring which is right Kasch, but not left Kasch. • Let S be the ring of power series on two noncommuting variables X and Y with coefficients from a field F. Let the ideal A be the ideal generated by the two elements YX and Y 2 . The quotient ring S/A is a local ring which is right Kasch but not left Kasch. • Suppose R is a ring direct product of infinitely many nonzero rings labeled A . The direct sum of the A forms a proper ideal of R. It is easily checked that the left and right annihilators of this ideal are zero, and so R is not right or left Kasch. • The two-by-two upper (or lower) triangular matrix ring is not right or left Kasch. • A ring with right socle zero (i.e. soc(RR ) = {0} ) cannot be right Kasch, since the ring contains no minimal right ideals. So, for example, domains which are not division rings are not right or left Kasch.

136.3 References [1] This ideal is necessarily a minimal right ideal.

• Faith, Carl (1999), Rings and things and a fine array of twentieth century associative algebra, Mathematical Surveys and Monographs 65, Providence, RI: American Mathematical Society, pp. xxxiv+422, ISBN 0-82180993-8, MR 1657671 • Kasch, Friedrich (1954), “Grundlagen einer Theorie der Frobeniuserweiterungen”, Math. Ann. (in German) 127: 453–474, doi:10.1007/bf01361137, ISSN 0025-5831, MR 0062724 • Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294 • Morita, Kiiti (1966), “On S-rings in the sense of F. Kasch”, Nagoya Math. J. 27: 687–695, ISSN 0027-7630, MR 0199230 • Nicholson, W. K.; Yousif, M. F. (2003), Quasi-Frobenius rings, Cambridge Tracts in Mathematics 158, Cambridge: Cambridge University Press, pp. xviii+307, doi:10.1017/CBO9780511546525, ISBN 0-521-81593-2, MR 2003785

Chapter 137

Krull ring In commutative algebra, a Krull ring or Krull domain is a commutative ring with a well behaved theory of prime factorization. They were introduced by Wolfgang Krull (1931). They are a higher-dimensional generalization of Dedekind domains, which are exactly the Krull domains of dimension at most 1. In this article, a ring is commutative and has unity.

137.1 Formal definition Let A be an integral domain and let P be the set of all prime ideals of A of height one, that is, the set of all prime ideals properly containing no nonzero prime ideal. Then A is a Krull ring if 1. Ap is a discrete valuation ring for all p ∈ P , 2. A is the intersection of these discrete valuation rings (considered as subrings of the quotient field of A ). 3. Any nonzero element of A is contained in only a finite number of height 1 prime ideals.

137.2 Properties A Krull domain is a unique factorization domain if and only if every prime ideal of height one is principal.[1] b is a Krull domain, then A is a Krull Let A be a Zariski ring (e.g., a local noetherian ring). If the completion A [2] domain.

137.3 Examples 1. Every integrally closed noetherian domain is a Krull ring. In particular, Dedekind domains are Krull rings. Conversely Krull rings are integrally closed, so a Noetherian domain is Krull if and only if it is integrally closed. 2. If A is a Krull ring then so is the polynomial ring A[x] and the formal power series ring A[[x]] . 3. The polynomial ring R[x1 , x2 , x3 , . . .] in infinitely many variables over a unique factorization domain R is a Krull ring which is not noetherian. In general, any unique factorization domain is a Krull ring. 4. Let A be a Noetherian domain with quotient field K , and L be a finite algebraic extension of K . Then the integral closure of A in L is a Krull ring (Mori–Nagata theorem).[3] 431

432

CHAPTER 137. KRULL RING

137.4 The divisor class group of a Krull ring A (Weil) divisor of a Krull ring A is a formal integral linear combination of the height 1 prime ideals, and these form a group D(A). A divisor of the form div(x) for some non-zero x in A is called a principal divisor, and the principal divisors form a subgroup of the group of divisors. The quotient of the group of divisors by the subgroup of principal divisors is called the divisor class group of A. A Cartier divisor of a Krull ring is a locally principal (Weil) divisor. The Cartier divisors form a subgroup of the group of divisors containing the principal divisors. The quotient of the Cartier divisors by the principal divisors is a subgroup of the divisor class group, isomorphic to the Picard group of invertible sheaves on Spec(A). Example: in the ring k[x,y,z]/(xy–z2 ] the divisor class group has order 2, generated by the divisor y=z, but the Picard subgroup is the trivial group.

137.5 References [1] http://eom.springer.de/k/k055930.htm [2] Bourbaki, 7.1, no 10, Proposition 16. [3] http://books.google.com/books?id=APPtnn84FMIC&lpg=PA83&ots=2L9MiWbIYZ&dq=krull%20akizuki&pg=PA85# v=onepage&q=krull%20akizuki&f=false

• N. Bourbaki. Commutative algebra. • Hazewinkel, Michiel, ed. (2001), “Krull ring”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608010-4 • Krull, Wolfgang (1931), “Allgemeine Bewertungstheorie”, J. Reine Angew. Math. 167: 160–196 • Hideyuki Matsumura, Commutative Algebra. Second Edition. Mathematics Lecture Note Series, 56. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. xv+313 pp. ISBN 0-8053-7026-9 • Hideyuki Matsumura, Commutative Ring Theory. Translated from the Japanese by M. Reid. Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge, 1986. xiv+320 pp. ISBN 0-52125916-9 • Samuel, Pierre (1964), Murthy, M. Pavman, ed., Lectures on unique factorization domains, Tata Institute of Fundamental Research Lectures on Mathematics 30, Bombay: Tata Institute of Fundamental Research, MR 0214579

Chapter 138

Krull’s principal ideal theorem In commutative algebra, Krull’s principal ideal theorem, named after Wolfgang Krull (1899–1971), gives a bound on the height of a principal ideal in a Noetherian ring. The theorem is sometimes referred to by its German name, Krulls Hauptidealsatz (Satz meaning “proposition” or “theorem”). Formally, if R is a Noetherian ring and I is a principal, proper ideal of R, then I has height at most one. This theorem can be generalized to ideals that are not principal, and the result is often called Krull’s height theorem. This says that if R is a Noetherian ring and I is a proper ideal generated by n elements of R, then I has height at most n. The principal ideal theorem and the generalization, the height theorem, both follow from the fundamental theorem of dimension theory. Bourbaki’s Commutative Algebra gives a direct proof. Kaplansky’s Commutative ring includes a proof due to David Rees.

138.1 References • Matsumura, Hideyuki (1970), Commutative Algebra, New York: Benjamin, see in particular section (12.I), p. 77 • http://www.math.lsa.umich.edu/~{}hochster/615W10/supDim.pdf

433

Chapter 139

Krull’s theorem In mathematics, and more specifically in ring theory, Krull’s theorem, named after Wolfgang Krull, asserts that a nonzero ring[1] has at least one maximal ideal. The theorem was proved in 1929 by Krull, who used transfinite induction. The theorem admits a simple proof using Zorn’s lemma, and in fact is equivalent to Zorn’s lemma, which in turn is equivalent to the axiom of choice.

139.1 Variants • For noncommutative rings, the analogues for maximal left ideals and maximal right ideals also hold. • For pseudo-rings, the theorem holds for regular ideals. • A slightly stronger (but equivalent) result, which can be proved in a similar fashion, is as follows: Let R be a ring, and let I be a proper ideal of R. Then there is a maximal ideal of R containing I. This result implies the original theorem, by taking I to be the zero ideal (0). Conversely, applying the original theorem to R/I leads to this result. To prove the stronger result directly, consider the set S of all proper ideals of R containing I. The set S is nonempty since I ∈ S. Furthermore, for any chain T of S, the union of the ideals in T is an ideal J, and a union of ideals not containing 1 does not contain 1, so J ∈ S. By Zorn’s lemma, S has a maximal element M. This M is a maximal ideal containing I.

139.2 Krull’s Hauptidealsatz Main article: Krull’s principal ideal theorem Another theorem commonly referred to as Krull’s theorem: Let R be a Noetherian ring and a an element of R which is neither a zero divisor nor a unit. Then every minimal prime ideal P containing a has height 1.

139.3 Notes [1] In this article, rings have a 1.

434

139.4. REFERENCES

435

139.4 References • W. Krull, Idealtheorie in Ringen ohne Endlichkeitsbedingungen, Mathematische Annalen 10 (1929), 729–744.

Chapter 140

Krull–Schmidt theorem In mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.

140.1 Definitions We say that a group G satisfies the ascending chain condition (ACC) on subgroups if every sequence of subgroups of G: 1 = G0 ≤ G1 ≤ G2 ≤ · · · is eventually constant, i.e., there exists N such that GN = GN₊₁ = GN₊₂ = ... . We say that G satisfies the ACC on normal subgroups if every such sequence of normal subgroups of G eventually becomes constant. Likewise, one can define the descending chain condition on (normal) subgroups, by looking at all decreasing sequences of (normal) subgroups: G = G0 ≥ G1 ≥ G2 ≥ · · · . Clearly, all finite groups satisfy both ACC and DCC on subgroups. The infinite cyclic group Z satisfies ACC but not DCC, since (2) > (2)2 > (2)3 > ... is an infinite decreasing sequence of subgroups. On the other hand, the p∞ -torsion part of Q/Z (the quasicyclic p-group) satisfies DCC but not ACC. We say a group G is indecomposable if it cannot be written as a direct product of non-trivial subgroups G = H × K.

140.2 Statement of the Theorem If G is a group that satisfies ACC and DCC on normal subgroups, then there is a unique way of writing G as a direct product G1 × G2 × · · · × Gk of finitely many indecomposable subgroups of G . Here, uniqueness means direct decompositions into indecomposable subgroups have the exchange property. That is: suppose G = H1 × H2 × · · · × Hl is another expression of G as a product of indecomposable subgroups. Then k = l and there is a reindexing of the Hi 's satisfying • Gi and Hi are isomorphic for each i ; • G = G1 × · · · × Gr × Hr+1 × · · · × Hl for each r .

140.3 Proof The proof is quite long and requires a sequence of technical lemmas, see the attached google book link [1] for details. 436

140.4. REMARK

437

140.4 Remark The theorem does not assert the existence of a non-trivial decomposition, but merely any such two decompositions (if exist) are same.

140.5 Krull–Schmidt Theorem for Modules If E ̸= 0 is a module that satisfies the ACC and DCC on submodules (that is, it is both Noetherian and Artinian or – equivalently – of finite length), then E is a direct sum of indecomposable modules. Up to a permutation, the indecomposable components in such a direct sum are uniquely determined up to isomorphism.[2] In general, the theorem fails if one only assumes that the module is Noetherian or Artinian.[3]

140.6 History The present-day Krull–Schmidt theorem was first proved by Joseph Wedderburn (Ann. of Math (1909)), for finite groups, though he mentions some credit is due to an earlier study of G.A. Miller where direct products of abelian groups were considered. Wedderburn’s theorem is stated as an exchange property between direct decompositions of maximum length. However, Wedderburn’s proof makes no use of automorphisms. The thesis of Robert Remak (1911) derived the same uniqueness result as Wedderburn but also proved (in modern terminology) that the group of central automorphisms acts transitively on the set of direct decompositions of maximum length of a finite group. From that stronger theorem Remak also proved various corollaries including that groups with a trivial center and perfect groups have a unique Remak decomposition. Otto Schmidt (Sur les produits directs, S. M. F. Bull. 41 (1913), 161–164), simplified the main theorems of Remak to the 3 page predecessor to today’s textbook proofs. His method improves Remak’s use of idempotents to create the appropriate central automorphisms. Both Remak and Schmidt published subsequent proofs and corollaries to their theorems. Wolfgang Krull (Über verallgemeinerte endliche Abelsche Gruppen, M. Z. 23 (1925) 161–196), returned to G.A. Miller's original problem of direct products of abelian groups by extending to abelian operator groups with ascending and descending chain conditions. This is most often stated in the language of modules. His proof observes that the idempotents used in the proofs of Remak and Schmidt can be restricted to module homomorphisms; the remaining details of the proof are largely unchanged. O. Ore unified the proofs from various categories include finite groups, abelian operator groups, rings and algebras by proving the exchange theorem of Wedderburn holds for modular lattices with descending and ascending chain conditions. This proof makes no use of idempotents and does not reprove the transitivity of Remak’s theorems. Kurosh’s The Theory of Groups and Zassenhaus’ The Theory of Groups include the proofs of Schmidt and Ore under the name of Remak–Schmidt but acknowledge Wedderburn and Ore. Later texts use the title Krull–Schmidt (Hungerford's Algebra) and Krull–Schmidt–Azumaya (Curtis–Reiner). The name Krull–Schmidt is now popularly substituted for any theorem concerning uniqueness of direct products of maximum size. Some authors choose to call direct decompositions of maximum-size Remak decompositions to honor his contributions.

140.7 See also • Krull–Schmidt category

140.8 References [1] Thomas W. Hungerford (6 December 2012). Algebra. Springer Science & Business Media. p. 83. ISBN 978-1-46126101-8. [2] Jacobson, Nathan (2009). Basic algebra 2 (2nd ed.). Dover. p. 115. ISBN 978-0-486-47187-7.

438

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[3] Facchini, Alberto; Herbera, Dolors; Levy, Lawrence S.; Vámos, Peter (1 December 1995). “Krull-Schmidt fails for Artinian modules”. Proceedings of the American Mathematical Society 123 (12): 3587–3587. doi:10.1090/S0002-99391995-1277109-4.

140.9 Further reading • A. Facchini: Module theory. Endomorphism rings and direct sum decompositions in some classes of modules. Progress in Mathematics, 167. Birkhäuser Verlag, Basel, 1998. ISBN 3-7643-5908-0 • C.M. Ringel: Krull–Remak–Schmidt fails for Artinian modules over local rings. Algebr. Represent. Theory 4 (2001), no. 1, 77–86.

140.10 External links • Page at PlanetMath

Chapter 141

Kummer ring In abstract algebra, a Kummer ring Z[ζ] is a subring of the ring of complex numbers, such that each of its elements has the form

n0 + n1 ζ + n2 ζ 2 + ... + nm−1 ζ m−1 where ζ is an mth root of unity, i.e.

ζ = e2πi/m and n0 through nm₋₁ are integers. A Kummer ring is an extension of Z , the ring of integers, hence the symbol Z[ζ] . Since the minimal polynomial of ζ is the mth cyclotomic polynomial, the ring Z[ζ] is an extension of degree ϕ(m) (where φ denotes Euler’s totient function). An attempt to visualize a Kummer ring on an Argand diagram might yield something resembling a quaint Renaissance map with compass roses and rhumb lines. The set of units of a Kummer ring contains {1, ζ, ζ 2 , . . . , ζ m−1 } . By Dirichlet’s unit theorem, there are also units of infinite order, except in the cases m = 1, m = 2 (in which case we have the ordinary ring of integers), the case m = 4 (the Gaussian integers) and the cases m = 3, m = 6 (the Eisenstein integers). Kummer rings are named after Ernst Kummer, who studied the unique factorization of their elements.

141.1 See also • Kummer theory

141.2 References • Allan Clark Elements of Abstract Algebra (1984 Courier Dover) p. 149

439

Chapter 142

Kurosh problem In mathematics, the Kurosh problem is one general problem, and several more special questions, in ring theory. The general problem is known to have a negative solution, since one of the special cases has been shown to have counterexamples. These matters were brought up by Aleksandr Gennadievich Kurosh as analogues of the Burnside problem in group theory. Kurosh asked whether there can be a finitely-generated infinite-dimensional algebraic algebra (the problem being to show this cannot happen). A special case is whether or not every nil algebra is locally nilpotent. For PI-algebras the Kurosh problem has a positive solution. Golod showed a counterexample to that case, as an application of the Golod–Shafarevich theorem. The Kurosh problem on group algebras concerns the augmentation ideal I. If I is a nil ideal, is the group algebra locally nilpotent?

142.1 References • Vesselin S. Drensky, Edward Formanek (2004), Polynomial Identity Rings, p. 89. • Some Open Problems in the Theory of Infinite Dimensional Algebras (PDF)

440

Chapter 143

Köthe conjecture In mathematics, the Köthe conjecture is a problem in ring theory, open as of 2010. It is formulated in various ways. Suppose that R is a ring. One way to state the conjecture is that if R has no nil ideal, other than {0}, then it has no nil one-sided ideal, other than {0}. This question was posed in 1930 by Gottfried Köthe (1905–1989). The Köthe conjecture has been shown to be true for various classes of rings, such as polynomial identity rings[1] and right Noetherian rings,[2] but a general solution remains elusive.

143.1 Equivalent formulations The conjecture has several different formulations:[3][4][5] 1. (Köthe conjecture) In any ring, the sum of two nil left ideals is nil. 2. In any ring, the sum of two one-sided nil ideals is nil. 3. In any ring, every nil left or right ideal of the ring is contained in the upper nil radical of the ring. 4. For any ring R and for any nil ideal J of R, then the matrix ideal Mn(J) is a nil ideal of Mn(R) for every n. 5. For any ring R and for any nil ideal J of R, then the matrix ideal M2(J) is a nil ideal of M2(R). 6. For any ring R, the upper nilradical of Mn(R) is the set of matrices with entries from the upper nilradical of R for every positive integer n. 7. For any ring R and for any nil ideal J of R, the polynomials with indeterminate x and coefficients from J lie in the Jacobson radical of the polynomial ring R[x]. 8. For any ring R, the Jacobson radical of R[x] consists of the polynomials with coefficients from the upper nilradical of R.

143.2 Related problems A conjecture by Amitsur read: “If J is a nil ideal in R, then J[x] is a nil ideal of the polynomial ring R[x].”[6] This conjecture, if true, would have proven the Köthe conjecture through the equivalent statements above, however a counterexample was produced by Agata Smoktunowicz.[7] While not a disproof of the Köthe conjecture, this fueled suspicions that the Köthe conjecture may be false in general.[8] In (Kegel 1962), it was proven that a ring which is the direct sum of two nilpotent subrings is itself nilpotent. The question arose whether or not “nilpotent” could be replaced with “locally nilpotent” or “nil”. Partial progress was made when Kelarev[9] produced an example of a ring which isn't nil, but is the direct sum of two locally nilpotent rings. This demonstrates that Kegel’s question with “locally nilpotent” replacing “nilpotent” is answered in the negative. The sum of a nilpotent subring and a nil subring is always nil.[10] 441

442

CHAPTER 143. KÖTHE CONJECTURE

143.3 References • Köthe, Gottfried (1930), “Die Struktur der Ringe, deren Restklassenring nach dem Radikal vollständig reduzibel ist”, Mathematische Zeitschrift 32 (1): 161–186, doi:10.1007/BF01194626 [1] John C. McConnell, James Christopher Robson, Lance W. Small , Noncommutative Noetherian rings (2001), p. 484. [2] Lam, T.Y., A First Course in Noncommutative Rings (2001), p.164. [3] Krempa, J., “Logical connections between some open problems concerning nil rings,” Fundamenta Mathematicae 76 (1972), no. 2, 121–130. [4] Lam, T.Y., A First Course in Noncommutative Rings (2001), p.171. [5] Lam, T.Y., Exercises in Classical Ring Theory (2003), p. 160. [6] Amitsur, S. A. Nil radicals. Historical notes and some new results Rings, modules and radicals (Proc. Internat. Colloq., Keszthely, 1971), pp. 47–65. Colloq. Math. Soc. János Bolyai, Vol. 6, North-Holland, Amsterdam, 1973. [7] Smoktunowicz, Agata. Polynomial rings over nil rings need not be nil J. Algebra 233 (2000), no. 2, p. 427–436. [8] Lam, T.Y., A First Course in Noncommutative Rings (2001), p.171. [9] Kelarev, A. V., A sum of two locally nilpotent rings may not be nil, Arch. Math. 60 (1993), p431–435. [10] Ferrero, M., Puczylowski, E. R., On rings which are sums of two subrings, Arch. Math. 53 (1989), p4–10.

143.4 External links • PlanetMath page • Survey paper (PDF)

Chapter 144

Lattice (module) In mathematics, in the field of ring theory, a lattice is a module over a ring which is embedded in a vector space over a field, giving an algebraic generalisation of the way a lattice group is embedded in a real vector space.

144.1 Formal definition Let R be an integral domain with field of fractions K. An R-module M is a lattice in the K-vector space V if M is finitely generated, R-torsion-free (no non-zero element of M is annihilated by a regular element of R) and an R-submodule of V. It is full if V = K·M.[1]

144.2 Pure sublattices A submodule N of M which is again a lattice is an R-pure sublattice if M/N is R-torsionfree. There is a one-to-one correspondence between R-pure sublattices N of M and K-subspaces W of V given by[2]

N 7→ W = K · N ; W 7→ N = W ∩ M.

144.3 See also • Lattice (group) for the case where M is a Z-module embedded in a vector space V over the field of real numbers R, and the Euclidean metric is used to describe the lattice structure

144.4 References [1] Reiner (2003) pp. 44, 108 [2] Reiner (2003) p. 45

• Reiner, I. (2003). Maximal Orders. London Mathematical Society Monographs. New Series 28. Oxford University Press. ISBN 0-19-852673-3. Zbl 1024.16008.

443

Chapter 145

Laurent polynomial In mathematics, a Laurent polynomial (named after Pierre Alphonse Laurent) in one variable over a field F is a linear combination of positive and negative powers of the variable with coefficients in F . Laurent polynomials in X form a ring denoted F [X, X−1 ].[1] They differ from ordinary polynomials in that they may have terms of negative degree. The construction of Laurent polynomials may be iterated, leading to the ring of Laurent polynomials in several variables.

145.1 Definition A Laurent polynomial with coefficients in a field F is an expression of the form

p=

∑

pk X k ,

pk ∈ F

k

where X is a formal variable, the summation index k is an integer (not necessarily positive) and only finitely many coefficients pk are non-zero. Two Laurent polynomials are equal if their coefficients are equal. Such expressions can be added, multiplied, and brought back to the same form by reducing similar terms. Formulas for addition and multiplication are exactly the same as for the ordinary polynomials, with the only difference that both positive and negative powers of X can be present: (

∑

) ai X

i

+

( ∑

i

) bi X

i

i

=

∑ (ai + bi )X i i

and (

∑ i

  )  ∑ ∑ ∑  ai X i ·  bj X j  = j

k

 ai bj  X k .

i,j:i+j=k

Since only finitely many coefficients ai and bj are non-zero, all sums in effect have only finitely many terms, and hence represent Laurent polynomials.

145.2 Properties • A Laurent polynomial over C may be viewed as a Laurent series in which only finitely many coefficients are non-zero. 444

145.3. SEE ALSO

445

• The ring of Laurent polynomials R[X, X−1 ] is an extension of the polynomial ring R[X] obtained by “inverting X". More rigorously, it is the localization of the polynomial ring in the multiplicative set consisting of the non-negative powers of X. Many properties of the Laurent polynomial ring follow from the general properties of localization. • The ring of Laurent polynomials is a subring of the rational functions. • The ring of Laurent polynomials over a field is Noetherian (but not Artinian). • If R is an integral domain, the units of the Laurent polynomial ring R[X, X−1 ] have the form uXk , where u is a unit of R and k is an integer. In particular, if K is a field then the units of K[X, X−1 ] have the form aXk , where a is a non-zero element of K. • The Laurent polynomial ring R[X, X−1 ] is isomorphic to the group ring of the group Z of integers over R. More generally, the Laurent polynomial ring in n variables is isomorphic to the group ring of the free abelian group of rank n. It follows that the Laurent polynomial ring can be endowed with a structure of a commutative, cocommutative Hopf algebra.

145.3 See also • Jones polynomial

145.4 References [1] Weisstein, Eric W., “Laurent Polynomial”, MathWorld.

Chapter 146

Length of a module In abstract algebra, the length of a module is a measure of the module’s “size”. It is defined to be the length of the longest chain of submodules and is a generalization of the concept of dimension for vector spaces. Modules with finite length share many important properties with finite-dimensional vector spaces. Other concepts used to 'count' in ring and module theory are depth and height; these are both somewhat more subtle to define. There are also various ideas of dimension that are useful. Finite length commutative rings play an essential role in functorial treatments of formal algebraic geometry.

146.1 Definition Let M be a (left or right) module over some ring R. Given a chain of submodules of M of the form

N0 ⊊ N1 ⊊ · · · ⊊ Nn we say that n is the length of the chain. The length of M is defined to be the largest length of any of its chains. If no such largest length exists, we say that M has infinite length. A ring R is said to have finite length as a ring if it has finite length as left R module.

146.2 Examples The zero module is the only one with length 0. Modules with length 1 are precisely the simple modules. For every finite-dimensional vector space (viewed as a module over the base field), the length and the dimension coincide. The length of the cyclic group Z/nZ (viewed as a module over the integers Z) is equal to the number of prime factors of n, with multiple prime factors counted multiple times.

146.3 Facts A module M has finite length if and only if it is both Artinian and Noetherian. (cf. Hopkins’ theorem) If M has finite length and N is a submodule of M, then N has finite length as well, and we have length(N) ≤ length(M). Furthermore, if N is a proper submodule of M (i.e. if it is unequal to M), then length(N) < length(M). If the modules M 1 and M 2 have finite length, then so does their direct sum, and the length of the direct sum equals the sum of the lengths of M 1 and M 2 . Suppose 446

146.4. SEE ALSO

447

0→L→M →N →0 is a short exact sequence of R-modules. Then M has finite length if and only if L and N have finite length, and we have length(M) = length(L) + length(N). (This statement implies the two previous ones.) A composition series of the module M is a chain of the form

0 = N0 ⊊ N1 ⊊ · · · ⊊ Nn = M such that

Ni+1 /Ni is simple for i = 0, . . . , n − 1 Every finite-length module M has a composition series, and the length of every such composition series is equal to the length of M.

146.4 See also • Hilbert–Poincaré series

146.5 References • Steven H. Weintraub, Representation Theory of Finite Groups AMS (2003) ISBN 0-8218-3222-0, ISBN 9780-8218-3222-6

Chapter 147

Levitzky’s theorem In mathematics, more specifically ring theory and the theory of nil ideals, Levitzky’s theorem, named after Jacob Levitzki, states that in a right Noetherian ring, every nil one-sided ideal is necessarily nilpotent.[1][2] Levitzky’s theorem is one of the many results suggesting the veracity of the Köthe conjecture, and indeed provided a solution to one of Köthe’s questions as described in (Levitzki 1945). The result was originally submitted in 1939 as (Levitzki 1950), and a particularly simple proof was given in (Utumi 1963).

147.1 Proof This is Utumi’s argument as it appears in (Lam 2001, p. 164-165) Lemma[3] Assume that R satisfies the ascending chain condition on annihilators of the form {r ∈ R | ar = 0} where a is in R. Then 1. Any nil one-sided ideal is contained in the lower nil radical Nil*(R); 2. Every nonzero nil right ideal contains a nonzero nilpotent right ideal. 3. Every nonzero nil left ideal contains a nonzero nilpotent left ideal. Levitzki’s Theorem [4] Let R be a right Noetherian ring. Then every nil one-sided ideal of R is nilpotent. In this case, the upper and lower nilradicals are equal, and moreover this ideal is the largest nilpotent ideal among nilpotent right ideals and among nilpotent left ideals. Proof: In view of the previous lemma, it is sufficient to show that the lower nilradical of R is nilpotent. Because R is right Noetherian, a maximal nilpotent ideal N exists. By maximality of N, the quotient ring R/N has no nonzero nilpotent ideals, so R/N is a semiprime ring. As a result, N contains the lower nilradical of R. Since the lower nilradical contains all nilpotent ideals, it also contains N, and so N is equal to the lower nilradical. Q.E.D.

147.2 See also • Nilpotent ideal • Köthe conjecture • Jacobson radical 448

147.3. NOTES

449

147.3 Notes [1] Herstein 1968, p. 37, Theorem 1.4.5 [2] Isaacs 1993, p. 210, Theorem 14.38 [3] Lam 2001, Lemma 10.29. [4] Lam 2001, Theorem 10.30.

147.4 References • Isaacs, I. Martin (1993), Algebra, a graduate course (1st ed.), Brooks/Cole Publishing Company, ISBN 0-53419002-2 • Herstein, I.N. (1968), Noncommutative rings (1st ed.), The Mathematical Association of America, ISBN 088385-015-X • Lam, T.Y. (2001), A First Course in Noncommutative Rings, Springer-Verlag, ISBN 978-0-387-95183-6 • Levitzki, J. (1950), “On multiplicative systems”, Compositio Mathematica 8: 76–80, MR 0033799. • Levitzki, Jakob (1945), “Solution of a problem of G. Koethe”, American Journal of Mathematics (The Johns Hopkins University Press) 67 (3): 437–442, doi:10.2307/2371958, ISSN 0002-9327, JSTOR 2371958, MR 0012269 • Utumi, Yuzo (1963), “Mathematical Notes: A Theorem of Levitzki”, The American Mathematical Monthly (Mathematical Association of America) 70 (3): 286, doi:10.2307/2313127, ISSN 0002-9890, JSTOR 2313127, MR 1532056

Chapter 148

Local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called “local behaviour”, in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies local rings and their modules. In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal. The concept of local rings was introduced by Wolfgang Krull in 1938 under the name Stellenringe.[1] The English term local ring is due to Zariski.[2]

148.1 Definition and first consequences A ring R is a local ring if it has any one of the following equivalent properties: • R has a unique maximal left ideal. • R has a unique maximal right ideal. • 1 ≠ 0 and the sum of any two non-units in R is a non-unit. • 1 ≠ 0 and if x is any element of R, then x or 1 − x is a unit. • If a finite sum is a unit, then it has a term that is a unit (this says in particular that the empty sum cannot be a unit, so it implies 1 ≠ 0). If these properties hold, then the unique maximal left ideal coincides with the unique maximal right ideal and with the ring’s Jacobson radical. The third of the properties listed above says that the set of non-units in a local ring forms a (proper) ideal,[3] necessarily contained in the Jacobson radical. The fourth property can be paraphrased as follows: a ring R is local if and only if there do not exist two coprime proper (principal) (left) ideals where two ideals I 1 , I 2 are called coprime if R = I 1 + I 2 . In the case of commutative rings, one does not have to distinguish between left, right and two-sided ideals: a commutative ring is local if and only if it has a unique maximal ideal. Before about 1960 many authors required that a local ring be (left and right) Noetherian, and (possibly non-Noetherian) local rings were called quasi-local rings. In this article this requirement is not imposed. A local ring that is an integral domain is called a local domain.

148.2 Examples • All fields (and skew fields) are local rings, since {0} is the only maximal ideal in these rings. • A nonzero ring in which every element is either a unit or nilpotent is a local ring. 450

148.2. EXAMPLES

451

• An important class of local rings are discrete valuation rings, which are local principal ideal domains that are not fields. • Every ring of formal power series F(X,Y,...) over a local ring F is local; the maximal ideal consists of those power series with constant term in the maximal ideal of the base ring. • Similarly, the algebra of dual numbers over any field is local. More generally, if F is a local ring and n is a positive integer, then the quotient ring F[X]/(Xn ) is local with maximal ideal consisting of the classes of polynomials with constant term belonging to the maximal ideal of F, since one can use a geometric series to invert all other polynomials modulo Xn . If F is a field, then elements of F[X]/(Xn ) are either nilpotent or invertible. (The dual numbers over F correspond to the case n=2.) • Quotient rings of local rings are local. • The ring of rational numbers with odd denominator is local; its maximal ideal consists of the fractions with even numerator and odd denominator: this is the integers localized at 2. • More generally, given any commutative ring R and any prime ideal P of R, the localization of R at P is local; the maximal ideal is the ideal generated by P in this localization.

148.2.1

Ring of germs

Main article: Germ (mathematics) To motivate the name “local” for these rings, we consider real-valued continuous functions defined on some open interval around 0 of the real line. We are only interested in the local behavior of these functions near 0 and we will therefore identify two functions if they agree on some (possibly very small) open interval around 0. This identification defines an equivalence relation, and the equivalence classes are the "germs of real-valued continuous functions at 0”. These germs can be added and multiplied and form a commutative ring. To see that this ring of germs is local, we need to identify its invertible elements. A germ f is invertible if and only if f(0) ≠ 0. The reason: if f(0) ≠ 0, then there is an open interval around 0 where f is non-zero, and we can form the function g(x) = 1/f(x) on this interval. The function g gives rise to a germ, and the product of fg is equal to 1. With this characterization, it is clear that the sum of any two non-invertible germs is again non-invertible, and we have a commutative local ring. The maximal ideal of this ring consists precisely of those germs f with f(0) = 0. Exactly the same arguments work for the ring of germs of continuous real-valued functions on any topological space at a given point, or the ring of germs of differentiable functions on any differentiable manifold at a given point, or the ring of germs of rational functions on any algebraic variety at a given point. All these rings are therefore local. These examples help to explain why schemes, the generalizations of varieties, are defined as special locally ringed spaces.

148.2.2

Valuation theory

Main article: Valuation (algebra) Local rings play a major role in valuation theory. By definition, a valuation ring of a field K is a subring R such that for every non-zero element x of K, at least one of x and x−1 is in R. Any such subring will be a local ring. For example, the ring of rational numbers with odd denominator (mentioned above) is a valuation ring in Q . Given a field K, which may or may not be a function field, we may look for local rings in it. If K were indeed the function field of an algebraic variety V, then for each point P of V we could try to define a valuation ring R of functions “defined at” P. In cases where V has dimension 2 or more there is a difficulty that is seen this way: if F and G are rational functions on V with F(P) = G(P) = 0, the function F/G

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is an indeterminate form at P. Considering a simple example, such as Y/X, approached along a line Y = tX, one sees that the value at P is a concept without a simple definition. It is replaced by using valuations.

148.2.3

Non-commutative

Non-commutative local rings arise naturally as endomorphism rings in the study of direct sum decompositions of modules over some other rings. Specifically, if the endomorphism ring of the module M is local, then M is indecomposable; conversely, if the module M has finite length and is indecomposable, then its endomorphism ring is local. If k is a field of characteristic p > 0 and G is a finite p-group, then the group algebra kG is local.

148.3 Some facts and definitions 148.3.1

Commutative Case

We also write (R, m) for a commutative local ring R with maximal ideal m. Every such ring becomes a topological ring in a natural way if one takes the powers of m as a neighborhood base of 0. This is the m-adic topology on R. If (R, m) and (S, n) are local rings, then a local ring homomorphism from R to S is a ring homomorphism f : R → S with the property f(m) ⊆ n. These are precisely the ring homomorphisms which are continuous with respect to the given topologies on R and S. A ring homomorphism f : R → S is a local ring homomorphism if and only if f −1 (n) = m ; that is, the preimage of the maximal ideal is maximal. As for any topological ring, one can ask whether (R, m) is complete (as a uniform space); if it is not, one considers its completion, again a local ring. If (R, m) is a commutative Noetherian local ring, then ∞ ∩

mi = {0}

i=1

(Krull’s intersection theorem), and it follows that R with the m-adic topology is a Hausdorff space. The theorem is a consequence of the Artin–Rees lemma, and, as such, the “Noetherian” assumption is crucial. Indeed, let R be the ring of germs of infinitely differentiable functions at 0 in the real line and m be the maximal ideal (x) . Then a 1 nonzero function e− x2 belongs to mn for any n, since that function divided by xn is still smooth. In algebraic geometry, especially when R is the local ring of a scheme at some point P, R / m is called the residue field of the local ring or residue field of the point P.

148.3.2

General Case

The Jacobson radical m of a local ring R (which is equal to the unique maximal left ideal and also to the unique maximal right ideal) consists precisely of the non-units of the ring; furthermore, it is the unique maximal two-sided ideal of R. However, in the non-commutative case, having a unique maximal two-sided ideal is not equivalent to being local.[4] For an element x of the local ring R, the following are equivalent:

148.4. NOTES

453

• x has a left inverse • x has a right inverse • x is invertible • x is not in m. If (R, m) is local, then the factor ring R/m is a skew field. If J ≠ R is any two-sided ideal in R, then the factor ring R/J is again local, with maximal ideal m/J. A deep theorem by Irving Kaplansky says that any projective module over a local ring is free, though the case where the module is finitely-generated is a simple corollary to Nakayama’s lemma. This has an interesting consequence in terms of Morita equivalence. Namely, if P is a finitely generated projective R module, then P is isomorphic to the free module Rn , and hence the ring of endomorphisms EndR (P ) is isomorphic to the full ring of matrices Mn (R) . Since every ring Morita equivalent to the local ring R is of the form EndR (P ) for such a P, the conclusion is that the only rings Morita equivalent to a local ring R are (isomorphic to) the matrix rings over R.

148.4 Notes [1] Krull, Wolfgang (1938). “Dimensionstheorie in Stellenringen”. J. Reine Angew. Math. (in German) 179: 204. [2] Zariski, Oscar (May 1943). “Foundations of a General Theory of Birational Correspondences”. Trans. Amer. Math. Soc. (American Mathematical Society) 53 (3): 490–542 [497]. doi:10.2307/1990215. JSTOR 1990215. [3] Lam (2001), p. 295, Thm. 19.1. [4] The 2 by 2 matrices over a field, for example, has unique maximal ideal {0}, but it has multiple maximal right and left ideals.

148.5 References • Lam, T.Y. (2001). A first course in noncommutative rings. Graduate Texts in Mathematics (2nd ed.). SpringerVerlag. ISBN 0-387-95183-0. • Jacobson, Nathan (2009). Basic algebra 2 (2nd ed.). Dover. ISBN 978-0-486-47187-7.

148.6 See also • Discrete valuation ring • Semi-local ring • Valuation ring

Chapter 149

Localization (algebra) In commutative algebra and algebraic geometry, the localization is a formal way to introduce the “denominators” to given a ring or a module. That is, it introduces a new ring/module out of an existing one so that it consists of fractions m s where the denominators s range in a given subset S of R. The basic example is the construction of the ring Q of rational numbers from the ring Z of rational integers. The technique has become fundamental, particularly in algebraic geometry, as it provides a natural link to sheaf theory. In fact, the term localization originates in algebraic geometry: if R is a ring of functions defined on some geometric object (algebraic variety) V, and one wants to study this variety “locally” near a point p, then one considers the set S of all functions which are not zero at p and localizes R with respect to S. The resulting ring R* contains only information about the behavior of V near p. Cf. the example given at local ring. An important related process is completion: one often localizes a ring/module, then completes. In this article, a ring is commutative with unity.

149.1 Construction 149.1.1

Localization of a ring

Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units (invertible elements) in R*. Further one wants R* to be the 'best possible' or 'most general' way to do this – in the usual fashion this should be expressed by a universal property. Let S be a multiplicatively closed subset of a ring R, i.e. for any s and t ∈ S, the product st is also in S, and 0 ̸∈ S and 1 ∈ S . Then the localization of R with respect to S, denoted S −1 R, is defined to be the following ring: as a set, it consists of equivalence classes of pairs (m, s), where m ∈ R and s ∈ S. Two such pairs (m, s) and (n, t) are considered equivalent if there is a third element u of S such that u(sn-tm) = 0 (The presence of u is crucial to the transitivity of ~) It is common to denote these equivalence classes m s Thus, S consists of “denominators”. To make this set a ring, define 454

149.1. CONSTRUCTION

455

tm + sn m n + := s t st and mn mn := s t st It is straightforward to check that the definition is well-defined, i.e. independent of choices of representatives of fractions. One then checks that the two operations are in fact addition and multiplication (associativity, etc) and that they are compatible (that is, distribution law). This step is also straightforward. The zero element is 0/1 and the unity is 1/1 ; they are usually simply denoted by 0 and 1. Finally, there is a canonical map j : R → S −1 R, m 7→ m/1 . (In general, it is not injective; if two elements of R differ by a nonzero zero-divisor with an annihilator in S, they have the same image by very definition.) The above mentioned universal property is the following: j : R → R* maps every element of S to a unit in R* (since (1/s)(s/1) = 1), and if f : R → T is some other ring homomorphism which maps every element of S to a unit in T, then there exists a unique ring homomorphism g : R* → T such that f = g ○ j If R has no nonzero zero-divisors (i.e., R is an integral domain), then the equivalence (m, s) ~ (n, t) reduces to sn = tm n which is precisely the condition we get when we formally clear out the denominators in m s = t . This motivates the definition above. In fact, the localization recovers the construction of the field of fractions as follows. Since the zero ideal is prime, its complement S is multiplicatively closed. The localization S −1 R then consists of r/s, r ∈ R, s ∈ R× . That is, S −1 R is precisely the field of fractions K of R. Since there is no nonzero zero-divisor, the canonical map m → m/1 is an inclusion and one can view R as a subring of K. Indeed, any localization of an integral domain is a subring of the field of fractions (cf. overring).

If S equals the complement of a prime ideal p ⊂ R (which is multiplicatively closed by definition of prime ideals), then the localization is denoted Rp. If S consists of all powers of a nonzero nilpotent f, then S −1 R is denoted by either Rf or R[f −1 ]. Another way to describe the localization of a ring R at a subset S is via category theory. If R is a ring and S is a subset, consider the set of all R-algebras A, so that, under the canonical homomorphism R → A, every element of S is mapped to a unit. The elements of this set form the objects of a category, with R-algebra homomorphisms as morphisms. Then, the localization of R at S is the initial object of this category.

149.1.2

Localization of a module

The construction above applies to a module M over a ring R except that instead of multiplication we define the scalar multiplication by

a·

m am := s s

Then S −1 M is a R -module consisting of m/s with the operations defined above. As above, there is a canonical module homomorphism φ: M → S −1 M mapping φ(m) = m / 1. The same notations for the localization of a ring are used for modules: Mp denote the localization of M at a prime ideal p and Mf the localization of a non-nilpotent element f. By the very definitions, the localization of the module is tightly linked to the one of the ring via the tensor product

456

CHAPTER 149. LOCALIZATION (ALGEBRA) S −1 M = M ⊗RS −1 R,

This way of thinking about localising is often referred to as extension of scalars. As a tensor product, the localization satisfies the usual universal property.

149.2 Examples and applications • Given a commutative ring R, we can consider the multiplicative set S of non-zerodivisors (i.e. elements a of R such that multiplication by a is an injection from R into itself.) The ring S −1 R is called the total quotient ring of R. S is the largest multiplicative set such that the canonical mapping from R to S −1 R is injective. When R is an integral domain, this is none other than the fraction field of R. • The ring Z/nZ where n is composite is not an integral domain. When n is a prime power it is a finite local ring, and its elements are either units or nilpotent. This implies it can be localized only to a zero ring. But when n can be factorised as ab with a and b coprime and greater than 1, then Z/nZ is by the Chinese remainder theorem isomorphic to Z/aZ × Z/bZ. If we take S to consist only of (1,0) and 1 = (1,1), then the corresponding localization is Z/aZ. • Let R = Z, and p a prime number. If S = Z - pZ, then R* is the localization of the integers at p. • As a generalization of the previous example, let R be a commutative ring and let p be a prime ideal of R. Then R - p is a multiplicative system and the corresponding localization is denoted Rp. The unique maximal ideal is then p. • Let R be a commutative ring and f an element of R. we can consider the multiplicative system {fn : n = 0,1,...}. Then the localization intuitively is just the ring obtained by inverting powers of f. If f is nilpotent, the localization is the zero ring. Two classes of localizations occur commonly in commutative algebra and algebraic geometry and are used to construct the rings of functions on open subsets in Zariski topology of the spectrum of a ring, Spec(R). • The set S consists of all powers of a given element r. The localization corresponds to restriction to the Zariski open subset Ur ⊂ Spec(R) where the function r is non-zero (the sets of this form are called principal Zariski open sets). For example, if R = K[X] is the polynomial ring and r = X then the localization produces the ring of Laurent polynomials K[X, X−1 ]. In this case, localization corresponds to the embedding U ⊂ A1 , where A1 is the affine line and U is its Zariski open subset which is the complement of 0. • The set S is the complement of a given prime ideal P in R. The primality of P implies that S is a multiplicatively closed set. In this case, one also speaks of the “localization at P". Localization corresponds to restriction to the complement U in Spec(R) of the irreducible Zariski closed subset V(P) defined by the prime ideal P.

149.3 Properties Some properties of the localization R* = S −1 R: • The ring homomorphism R → S −1 R is injective if and only if S does not contain any zero divisors. • There is a bijection between the set of prime ideals of S −1 R and the set of prime ideals of R which do not intersect S. This bijection is induced by the given homomorphism R → S −1 R. • In particular: after localization at a prime ideal P, one obtains a local ring, or in other words, a ring with one maximal ideal, namely the ideal generated by the extension of P. The localization of a module M → S −1 M is a functor from the category of R-modules to the category of S −1 R -modules. From the definition, one can see that it is exact, or in other words (reading this in the tensor product) that

149.4. STABILITY UNDER LOCALIZATION

457

S −1 R is a flat module over R. This is actually foundational for the use of flatness in algebraic geometry, saying in particular that the inclusion of an open set in Spec(R) (see spectrum of a ring) is a flat morphism. The localization functor (usually) preserves Hom and tensor products in the following sense: the natural map

S −1 (M ⊗R N ) → S −1 M ⊗S −1 R S −1 N is an isomorphism and if M is finitely presented, the natural map

S −1 HomR (M, N ) → HomS −1 R (S −1 M, S −1 N ) is an isomorphism. If a module M is a finitely generated over R, we have: S −1 M = 0 if and only if tM = 0 for some t ∈ S if and only if S intersects the annihilator of M.[1] Let R be an integral domain with the field of fractions K. Then its localization Rp at a prime ideal p can be viewed as a subring of K. Moreover,

R = ∩p Rp = ∩m Rm where the first intersection is over all prime ideals and the second over the maximal ideals.[2] √ Let I denote the radical of an ideal I in R. Then √ √ I · S −1 R = I · S −1 R In particular, R is reduced if and only if its total ring of fractions is reduced.[3]

149.4 Stability under localization Many properties of a ring are stable under localization. For example, the localization of a noetherian ring (resp. principal ideal domain) is noetherian (resp. principal ideal domain). The localization of an integrally closed domain is an integrally closed domain. In many cases, the converse also holds. (See below)

149.5 Local property Let M be a R-module. We could think of two kinds of what it means some property P holds for M at a prime ideal p . One means that P holds for Mp ; the other means that P holds for a neighborhood of p . The first interpretation is more common.[4] But for many properties the first and second interpretations coincide. Explicitly, the second means the following conditions are equivalent. • (i) P holds for M. • (ii) P holds for Mp for all prime ideal p of R. • (iii) P holds for Mm for all maximal ideal m of R. Then the following are local properties in the second sense. • M is zero. • M is torsion-free (when R is a domain) • M is flat.

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CHAPTER 149. LOCALIZATION (ALGEBRA)

• M is invertible module (when R is a domain and M is a submodule of the field of fractions of R) • f : M → N is injective (resp. surjective) when N is another R-module. On the other hand, some properties are not local properties. For example, “noetherian” is (in general) not a local property: that is, to say there is a non-noetherian ring whose localization at every maximal ideal is noetherian: this example is due to Nagata.

149.6 Support The support of the module M is the set of prime ideals p such that Mp ≠ 0. Viewing M as a function from the spectrum of R to R-modules, mapping

p 7→ Mp this corresponds to the support of a function.

149.7 (Quasi-)coherent sheaves In terms of localization of modules, one can define quasi-coherent sheaves and coherent sheaves on locally ringed spaces. In algebraic geometry, the quasi-coherent OX-modules for schemes X are those that are locally modelled on sheaves on Spec(R) of localizations of any R-module M. A coherent OX-module is such a sheaf, locally modelled on a finitely-presented module over R.

149.8 Non-commutative case Localizing non-commutative rings is more difficult; the localization does not exist for every set S of prospective units. One condition which ensures that the localization exists is the Ore condition. One case for non-commutative rings where localization has a clear interest is for rings of differential operators. It has the interpretation, for example, of adjoining a formal inverse D−1 for a differentiation operator D. This is done in many contexts in methods for differential equations. There is now a large mathematical theory about it, named microlocalization, connecting with numerous other branches. The micro- tag is to do with connections with Fourier theory, in particular.

149.9 See also • Completion (ring theory) • Valuation ring • Overring

149.9.1

Localization

Category:Localization (mathematics) • Local analysis • Localization of a category • Localization of a ring

149.10. NOTES • Localization of a module • Localization of a topological space • Local ring

149.10 Notes [1] Borel, AG. 3.1 [2] Matsumura, Theorem 4.7 [3] Borel, AG. 3.3 [4] Matsumura, a remark after Theorem 4.5

149.11 References • Borel, Armand. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag. ISBN 0-387-97370-2. • Serge Lang, “Algebraic Number Theory,” Springer, 2000. pages 3–4.

459

Chapter 150

Localization of a module In algebraic geometry, the localization of a module is a construction to introduce denominators in a module for a ring. More precisely, it is a systematic way to construct a new module S −1 M out of a given module M containing algebraic fractions m s where the denominators s range in a given subset S of R. The technique has become fundamental, particularly in algebraic geometry, as the link between modules and sheaf theory. Localization of a module generalizes localization of a ring.

150.1 Definition In this article, let R be a commutative ring and M an R-module. Let S a multiplicatively closed subset of R, i.e. 1 ∈ S and for any s and t ∈ S, the product st is also in S. Then the localization of M with respect to S, denoted S −1 M, is defined to be the following module: as a set, it consists of equivalence classes of pairs (m, s), where m ∈ M and s ∈ S. Two such pairs (m, s) and (n, t) are considered equivalent if there is a third element u of S such that u(sn-tm) = 0 It is common to denote these equivalence classes m s To make this set a R-module, define

m n tm + sn + := s t st and

a·

m am := s s

(a ∈ R). It is straightforward to check that the definition is well-defined, i.e. yields the same result for different choices of representatives of fractions. One interesting characterization of the equivalence relation is that it is the smallest 460

150.2. REMARKS

461

relation (considered as a set) such that cancellation laws hold for elements in S. That is, it is the smallest relation such that rs/us = r/u for all s in S. One case is particularly important: if S equals the complement of a prime ideal p ⊂ R (which is multiplicatively closed by definition of prime ideals) then the localization is denoted Mp instead of (R\p)−1 M. The support of the module M is the set of prime ideals p such that Mp ≠ 0. Viewing M as a function from the spectrum of R to R-modules, mapping

p 7→ Mp this corresponds to the support of a function. Localization of a module at primes also reflects the “local properties” of the module. In particular, there are many cases where the more general situation can be reduced to a statement about localized modules. The reduction is because a R-module M is trivial if and only if all its localizations at primes or maximal ideals are trivial.

150.2 Remarks • The definition applies in particular to M=R, and we get back the localized ring S −1 R. • There is a module homomorphism φ: M → S −1 M mapping φ(m) = m / 1. Here φ need not be injective, in general, because there may be significant torsion. The additional u showing up in the definition of the above equivalence relation can not be dropped (otherwise the relation would not be transitive), unless the module is torsion-free. • Some authors allow not necessarily multiplicatively closed sets S and define localizations in this situation, too. However, saturating such a set, i.e. adding 1 and finite products of all elements, this comes down to the above definition.

150.3 Tensor product interpretation By the very definitions, the localization of the module is tightly linked to the one of the ring via the tensor product S −1 M = M ⊗RS −1 R, This way of thinking about localising is often referred to as extension of scalars. As a tensor product, the localization satisfies the usual universal property.

150.4 Flatness From the definition, one can see that localization of modules is an exact functor, or in other words (reading this in the tensor product) that S −1 R is a flat module over R. This fact is foundational for the use of flatness in algebraic geometry, saying in particular that the inclusion of the open set Spec(S −1 R) into Spec(R) (see spectrum of a ring) is a flat morphism.

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CHAPTER 150. LOCALIZATION OF A MODULE

150.5 (Quasi-)coherent sheaves In terms of localization of modules, one can define quasi-coherent sheaves and coherent sheaves on locally ringed spaces. In algebraic geometry, the quasi-coherent OX-modules for schemes X are those that are locally modelled on sheaves on Spec(R) of localizations of any R-module M. A coherent OX-module is such a sheaf, locally modelled on a finitely-presented module over R.

150.6 See also 150.6.1

Localization

Category:Localization (mathematics) • Local analysis • Localization (algebra) • Localization of a category • Localization of a ring • Localization of a topological space

150.7 References Any textbook on commutative algebra covers this topic, such as: • Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics 150, Berlin, New York: SpringerVerlag, ISBN 978-0-387-94268-1, MR 1322960

Chapter 151

Localization of a ring In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units (invertible elements) in R*. Further one wants R* to be the 'best possible' or 'most general' way to do this – in the usual fashion this should be expressed by a universal property. The localization of R by S is usually denoted by S −1 R; however other notations are used in some important special cases. If S is the set of the non zero elements of an integral domain, then the localization is the field of fractions and thus usually denoted Frac(R). If S is the complement of a prime ideal I the localization is denoted by RI, and Rf is used to denote the localization by the powers of an element f.[1] The two latter cases are fundamental in algebraic geometry and scheme theory. In particular the definition of an affine scheme is based on the properties of these two kinds of localizations. An important related process is completion: one often localizes a ring, then completes.

151.1 Terminology The term localization originates in algebraic geometry: if R is a ring of functions defined on some geometric object (algebraic variety) V, and one wants to study this variety “locally” near a point p, then one considers the set S of all functions that are not zero at p and localizes R with respect to S. The resulting ring R* contains only information about the behavior of V near p. Cf. the example given at local ring. In number theory and algebraic topology, one refers to the behavior of a ring at a number n or away from n. “Away from n" means “in the ring localized by the set of the powers of n" (which is a Z[1/n]-algebra). If n is a prime number, “at n" means “in the ring localized by the set of the integers which are not multiple of n".

151.2 Construction and properties for commutative rings The set S is assumed to be a submonoid of the multiplicative monoid of R, i.e. 1 is in S and for s and t in S we also have st in S. A subset of R with this property is called a multiplicatively closed set or more briefly, a multiplicative set. This requirement on S is natural and necessary to have since its elements will be turned into units of the localization, and units must be closed under multiplication. It is standard practice to assume that S is multiplicatively closed. If S is not multiplicatively closed, it suffices to replace it by its multiplicative closure, consisting of the set of the products of elements of S (including the empty product 1). This does not change the result of the localization. The fact that we talk of “a localization with respect to the powers of an element” instead of “a localization with respect to an element” is an example of this. Therefore we shall suppose S to be multiplicatively closed in what follows.

151.2.1

Construction 463

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CHAPTER 151. LOCALIZATION OF A RING

For integral domains In case R is an integral domain there is an easy construction of the localization. Since the only ring in which 0 is a unit is the trivial ring {0}, the localization R* is {0} if 0 is in S. Otherwise, the field of fractions K of R can be used: we take R* to be the subset of K consisting of the elements of the form r/s with r in R and s in of S; as we have supposed S multiplicatively closed, R* is a subring. The standard embedding of R into R* is injective in this case, although it may be non injective in a more general setting. For example, the dyadic fractions are the localization of the ring of integers with respect to the powers of two. In this case, R* is the dyadic fractions, R is the integers, the denominators are powers of 2, and the natural map from R to R* is injective. The result would be exactly the same if we had taken S={2}. For general commutative rings For general commutative rings, we don't have a field of fractions. Nevertheless, a localization can be constructed consisting of “fractions” with denominators coming from S; in contrast with the integral domain case, one can safely 'cancel' from numerator and denominator only elements of S. This construction proceeds as follows: on R × S define an equivalence relation ~ by setting (r1 ,s1 ) ~ (r2 ,s2 ) if there exists t in S such that t(r1 s2 − r2 s1 ) = 0. (The presence of t is crucial to the transitivity of ~) We think of the equivalence class of (r,s) as the “fraction” r/s and, using this intuition, the set of equivalence classes R* can be turned into a ring with operations that look identical to those of elementary algebra: a/s + b/t = (at + bs)/st and (a/s)(b/t) = ab/st. The map j : R → R* that maps r to the equivalence class of (r,1) is then a ring homomorphism. In general, this is not injective; if a and b are two elements of R such that there exists s in S with s(a − b) = 0, then their images under j are equal. Universal property The above-mentioned universal property is the following: the ring homomorphism j : R → R* maps every element of S to a unit in R*, and if f : R → T is some other ring homomorphism which maps every element of S to a unit in T, then there exists a unique ring homomorphism g : R* → T such that f = g ∘ j.

151.2.2

Examples

• Given a commutative ring R, we can consider the multiplicative set S of non-zero-divisors (i.e. elements a of R such that multiplication by a is an injection from R into itself.) The ring S −1 R is called the total quotient ring of R. S is the largest multiplicative set such that the canonical mapping from R to S −1 R is injective. When R is an integral domain, this is the fraction field of R. • The ring Z/nZ where n is composite is not an integral domain. When n is a prime power it is a finite local ring, and its elements are either units or nilpotent. This implies it can be localized only to a zero ring. But when n can be factorised as ab with a and b coprime and greater than 1, then Z/nZ is by the Chinese remainder theorem isomorphic to Z/aZ × Z/bZ. If we take S to consist only of (1,0) and 1 = (1,1), then the corresponding localization is Z/aZ. • Let R = Z, and p a prime number. If S = Z - pZ, then R* is the localization of the integers at p. See Lang’s “Algebraic Number Theory,” especially pages 3–4 and the bottom of page 7. • As a generalization of the previous example, let R be a commutative ring and let p be a prime ideal of R. Then R - p is a multiplicative system and the corresponding localization is denoted Rp. The unique maximal ideal is then pRp. • Let R be a commutative ring and f an element of R. we can consider the multiplicative system {fn : n = 0,1,...}. Then the localization intuitively is just the ring obtained by inverting powers of f. If f is nilpotent, the localization is the zero ring.

151.3. CATEGORY THEORETIC DESCRIPTION

151.2.3

465

Properties

Some properties of the localization R* = S −1 R: • S −1 R = {0} if and only if S contains 0. • The ring homomorphism R → S −1 R is injective if and only if S does not contain any zero divisors. • There is a bijection between the set of prime ideals of S −1 R and the set of prime ideals of R which do not intersect S. This bijection is induced by the given homomorphism R → S −1 R. • In particular: after localization at a prime ideal P, one obtains a local ring, or in other words, a ring with one maximal ideal, namely the ideal generated by the extension of P.

151.3 Category theoretic description Another way to describe the localization of a ring R at a subset S is via category theory. If R is a ring and S is a subset, consider all R-algebras A, so that, under the canonical homomorphism R → A, every element of S is mapped to a unit. These algebras are the objects of a category, with R-algebra homomorphisms as morphisms. Then, the localization of R at S is the initial object of this category. (This is a more abstract way of expressing the universal property above.)

151.4 Applications Two classes of localizations occur commonly in commutative algebra and algebraic geometry and are used to construct the rings of functions on open subsets in Zariski topology of the spectrum of a ring, Spec(R). • The set S consists of all powers of a given element r. The localization corresponds to restriction to the Zariski open subset Ur ⊂ Spec(R) where the function r is non-zero (the sets of this form are called principal Zariski open sets). For example, if R = K[X] is the polynomial ring and r = X then the localization produces the ring of Laurent polynomials K[X, X−1 ]. In this case, localization corresponds to the embedding U ⊂ A1 , where A1 is the affine line and U is its Zariski open subset which is the complement of 0. • The set S is the complement of a given prime ideal P in R. The primality of P implies that S is a multiplicatively closed set. In this case, one also speaks of the “localization at P". Localization corresponds to restriction to arbitrary small open neighborhoods of the irreducible Zariski closed subset V(P) defined by the prime ideal P in Spec(R).

151.5 Non-commutative case Localizing non-commutative rings is more difficult; the localization does not exist for every set S of prospective units. One condition which ensures that the localization exists is the Ore condition. One case for non-commutative rings where localization has a clear interest is for rings of differential operators. It has the interpretation, for example, of adjoining a formal inverse D−1 for a differentiation operator D. This is done in many contexts in methods for differential equations. There is now a large mathematical theory about it, named microlocalization, connecting with numerous other branches. The micro- tag is to do with connections with Fourier theory, in particular.

151.6 See also • Completion (ring theory) • Homomorphism • Overring • Valuation ring

466

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Localization

Category:Localization (mathematics) • Local analysis • Local ring • Localization (algebra) • Localization of a category • Localization of a module • Localization of a topological space

151.7 References [1] Eisenbud, Harris, The geometry of schemes

• Cohn, P. M. (1989). "§ 9.3”. Algebra. Vol. 2 (2nd ed.). Chichester: John Wiley & Sons Ltd. pp. xvi+428. ISBN 0-471-92234-X. MR 1006872. • Cohn, P. M. (1991). "§ 9.1”. Algebra. Vol. 3 (2nd ed.). Chichester: John Wiley & Sons Ltd. pp. xii+474. ISBN 0-471-92840-2. MR 1098018. • Stenström, Bo (1971). Rings and modules of quotients. Lecture Notes in Mathematics, Vol. 237. Berlin: Springer-Verlag. pp. vii+136. ISBN 978-3-540-05690-4. MR 0325663. • Serge Lang, “Algebraic Number Theory,” Springer, 2000. pages 3–4.

151.8 External links • Localization from MathWorld.

Chapter 152

Loewy ring In mathematics, a Loewy ring or semi-Artinian ring is a ring in which every non-zero module has a non-zero socle, or equivalently if the Loewy length of every module is defined. The concepts are named after Alfred Loewy.

152.1 Loewy length The Loewy length and Loewy series were introduced by Emil Artin, Cecil J. Nesbitt, and Robert M. Thrall (1944) If M is a module, then define the Loewy series Mα for ordinals α by M 0 = 0, Mα₊₁/Mα = socle M/Mα, Mα = ∪λ 0, let n ⋅ x = x + x + ... + x (n summands), 0 ⋅ x = 0, and (−n) ⋅ x = −(n ⋅ x). Such a module need not have a basis—groups containing torsion elements do not. (For example, in the group of integers modulo 3, one cannot find even one element which satisfies the definition of a linearly independent set since when an integer such as 3 or 6 multiplies an element the result is 0. However if a finite field is considered as a module over the same finite field taken as a ring, it is a vector space and does have a basis.) • If R is any ring and n a natural number, then the cartesian product Rn is both a left and a right module over R if we use the component-wise operations. Hence when n = 1, R is an R-module, where the scalar multiplication is just ring multiplication. The case n = 0 yields the trivial R-module {0} consisting only of its identity element. Modules of this type are called free and if R has invariant basis number (e.g. any commutative ring or field) the number n is then the rank of the free module. • If S is a nonempty set, M is a left R-module, and M S is the collection of all functions f : S → M, then with addition and scalar multiplication in M S defined by (f + g)(s) = f(s) + g(s) and (rf)(s) = rf(s), M S is a left R-module. The right R-module case is analogous. In particular, if R is commutative then the collection of R-module homomorphisms h : M → N (see below) is an R-module (and in fact a submodule of N M ). • If X is a smooth manifold, then the smooth functions from X to the real numbers form a ring C ∞ (X). The set of all smooth vector fields defined on X form a module over C ∞ (X), and so do the tensor fields and the differential forms on X. More generally, the sections of any vector bundle form a projective module over C ∞ (X), and by Swan’s theorem, every projective module is isomorphic to the module of sections of some bundle; the category of C ∞ (X)-modules and the category of vector bundles over X are equivalent. • The square n-by-n matrices with real entries form a ring R, and the Euclidean space Rn is a left module over this ring if we define the module operation via matrix multiplication. • If R is any ring and I is any left ideal in R, then I is a left module over R. Analogously of course, right ideals are right modules.

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• If R is a ring, we can define the ring Rop which has the same underlying set and the same addition operation, but the opposite multiplication: if ab = c in R, then ba = c in Rop . Any left R-module M can then be seen to be a right module over Rop , and any right module over R can be considered a left module over Rop . • There are modules of a Lie algebra as well.

160.3 Submodules and homomorphisms Suppose M is a left R-module and N is a subgroup of M. Then N is a submodule (or R-submodule, to be more explicit) if, for any n in N and any r in R, the product r ⋅ n is in N (or n ⋅ r for a right module). The set of submodules of a given module M, together with the two binary operations + and ∩, forms a lattice which satisfies the modular law: Given submodules U, N 1 , N 2 of M such that N 1 ⊂ N 2 , then the following two submodules are equal: (N 1 + U) ∩ N 2 = N 1 + (U ∩ N 2 ). If M and N are left R-modules, then a map f : M → N is a homomorphism of R-modules if, for any m, n in M and r, s in R,

f (r · m + s · n) = r · f (m) + s · f (n) This, like any homomorphism of mathematical objects, is just a mapping which preserves the structure of the objects. Another name for a homomorphism of modules over R is an R-linear map. A bijective module homomorphism is an isomorphism of modules, and the two modules are called isomorphic. Two isomorphic modules are identical for all practical purposes, differing solely in the notation for their elements. The kernel of a module homomorphism f : M → N is the submodule of M consisting of all elements that are sent to zero by f. The isomorphism theorems familiar from groups and vector spaces are also valid for R-modules. The left R-modules, together with their module homomorphisms, form a category, written as R-Mod (see category of modules for more.) This is an abelian category.

160.4 Types of modules Finitely generated. An R-module M is finitely generated if there exist finitely many elements x1 , ..., xn in M such that every element of M is a linear combination of those elements with coefficients from the ring R. Cyclic. A module is called a cyclic module if it is generated by one element. Free. A free R-module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the ring R. These are the modules that behave very much like vector spaces. Projective. Projective modules are direct summands of free modules and share many of their desirable properties. Injective. Injective modules are defined dually to projective modules. Flat. A module is called flat if taking the tensor product of it with any exact sequence of R-modules preserves exactness. Torsionless module. A module is called torsionless if it embeds into its algebraic dual. Simple. A simple module S is a module that is not {0} and whose only submodules are {0} and S. Simple modules are sometimes called irreducible.[2] Semisimple. A semisimple module is a direct sum (finite or not) of simple modules. Historically these modules are also called completely reducible. Indecomposable. An indecomposable module is a non-zero module that cannot be written as a direct sum of two non-zero submodules. Every simple module is indecomposable, but there are indecomposable modules which are not simple (e.g. uniform modules). Faithful. A faithful module M is one where the action of each r ≠ 0 in R on M is nontrivial (i.e. r ⋅ x ≠ 0 for some x in M). Equivalently, the annihilator of M is the zero ideal.

160.5. FURTHER NOTIONS

489

Torsion-free. A torsion-free module is a module over a ring such that 0 is the only element annihilated by a regular element (non zero-divisor) of the ring. Noetherian. A Noetherian module is a module which satisfies the ascending chain condition on submodules, that is, every increasing chain of submodules becomes stationary after finitely many steps. Equivalently, every submodule is finitely generated. Artinian. An Artinian module is a module which satisfies the descending chain condition on submodules, that is, every decreasing chain of submodules becomes stationary after finitely many steps. Graded. A graded module is a module with a decomposition as a direct sum M = ⨁x Mx over a graded ring R = ⨁x Rx such that RxMy ⊂ Mx ₊ y for all x and y. Uniform. A uniform module is a module in which all pairs of nonzero submodules have nonzero intersection.

160.5 Further notions 160.5.1

Relation to representation theory

If M is a left R-module, then the action of an element r in R is defined to be the map M → M that sends each x to rx (or xr in the case of a right module), and is necessarily a group endomorphism of the abelian group (M,+). The set of all group endomorphisms of M is denoted EndZ(M) and forms a ring under addition and composition, and sending a ring element r of R to its action actually defines a ring homomorphism from R to EndZ(M). Such a ring homomorphism R → EndZ(M) is called a representation of R over the abelian group M; an alternative and equivalent way of defining left R-modules is to say that a left R-module is an abelian group M together with a representation of R over it. A representation is called faithful if and only if the map R → EndZ(M) is injective. In terms of modules, this means that if r is an element of R such that rx = 0 for all x in M, then r = 0. Every abelian group is a faithful module over the integers or over some modular arithmetic Z/nZ.

160.5.2

Generalizations

Any ring R can be viewed as a preadditive category with a single object. With this understanding, a left R-module is nothing but a (covariant) additive functor from R to the category Ab of abelian groups. Right R-modules are contravariant additive functors. This suggests that, if C is any preadditive category, a covariant additive functor from C to Ab should be considered a generalized left module over C; these functors form a functor category C-Mod which is the natural generalization of the module category R-Mod. Modules over commutative rings can be generalized in a different direction: take a ringed space (X, OX) and consider the sheaves of OX-modules; see sheaf of modules for more. These form a category OX-Mod, and play an important role in modern algebraic geometry. If X has only a single point, then this is a module category in the old sense over the commutative ring OX(X). One can also consider modules over a semiring. Modules over rings are abelian groups, but modules over semirings are only commutative monoids. Most applications of modules are still possible. In particular, for any semiring S the matrices over S form a semiring over which the tuples of elements from S are a module (in this generalized sense only). This allows a further generalization of the concept of vector space incorporating the semirings from theoretical computer science. Over near-rings, one can consider near-ring modules, a nonabelian generalization of modules.

160.6 See also • group ring • algebra (ring theory) • module (model theory)

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• module spectrum

160.7 Notes [1] This is the endomorphism ring of the additive group M. If R is commutative, then these endomorphisms are additionally R linear. [2] Jacobson (1964), p. 4, Def. 1; Irreducible Module at PlanetMath.org.

160.8 References • F.W. Anderson and K.R. Fuller: Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13, 2nd Ed., Springer-Verlag, New York, 1992, ISBN 0-387-97845-3, ISBN 3-540-97845-3 • Nathan Jacobson. Structure of rings. Colloquium publications, Vol. 37, 2nd Ed., AMS Bookstore, 1964, ISBN 978-0-8218-1037-8

160.9 External links • Hazewinkel, Michiel, ed. (2001), “Module”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-0104 • Why is it a good idea to study the modules of a ring ? on MathOverflow • module in nLab

Chapter 161

Module of covariants In algebra, given an algebraic group G, a G-module M and a G-algebra A, all over a field k, the module of covariants of type M is the AG -module

(M ⊗k A)G .

161.1 See also • Local cohomology

161.2 References • M. Brion, Sur les modules de covariants, Ann. Sci. École Norm. Sup. (4) 26 (1993), 1 21. • M. Van den Bergh, Modules of covarariants, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zurich, 1994), Birkhauser, Basel, pp. 352-362, 1995.

491

Chapter 162

Monoid ring In abstract algebra, a monoid ring is a ring constructed from a ring and a monoid, just as a group ring is constructed from a ring and a group.

162.1 Definition Let R be a ring and let ∑ G be a monoid. The monoid ring or monoid algebra of G over R, denoted R[G] or RG, is the set of formal sums g∈G rg g , where rg ∈ R for each g ∈ G and rg = 0 for all but finitely many g, equipped with coefficient-wise addition, and the multiplication in which the elements of R commute with the elements of G. More formally, R[G] is the set of functions φ: G → R such that {g : φ(g) ≠ 0} is finite, equipped with addition of functions, and with multiplication defined by

(ϕψ)(g) =

∑

ϕ(k)ψ(ℓ)

kℓ=g

If G is a group, then R[G] is also called the group ring of G over R.

162.2 Universal property Given R and G, there is a ring homomorphism α: R → R[G] sending each r to r1 (where 1 is the identity element of G), and a monoid homomorphism β: G → R[G] (where the latter is viewed as a monoid under multiplication) sending each g to 1g (where 1 is the multiplicative identity of R). We have that α(r) commutes with β(g) for all r in R and g in G. The universal property of the monoid ring states that given a ring S, a ring homomorphism α': R → S, and a monoid homomorphism β': G → S to the multiplicative monoid of S, such that α'(r) commutes with β'(g) for all r in R and g in G, there is a unique ring homomorphism γ: R[G] → S such that composing α and β with γ produces α' and β '.

162.3 Augmentation The augmentation is the ring homomorphism η: R[G] → R defined by

η(

∑ g∈G

rg g) =

∑

rg .

g∈G

The kernel of η is called the augmentation ideal. It is a free R-module with basis consisting of 1–g for all g in G not equal to 1. 492

162.4. EXAMPLES

493

162.4 Examples Given a ring R and the (additive) monoid of natural numbers N (or {xn } viewed multiplicatively), we obtain the ring R[{xn }] =: R[x] of polynomials over R. The monoid Nn (with the addition) gives the polynomial ring with n variables: R[Nn ] =: R[X1 , ..., Xn].

162.5 Generalization If G is a semigroup, the same construction yields a semigroup ring R[G].

162.6 See also • Free algebra

162.7 References • Lang, Serge (2002). Algebra. Graduate Texts in Mathematics 211 (Rev. 3rd ed.). New York: Springer-Verlag. ISBN 0-387-95385-X.

162.8 Further reading • R.Gilmer. Commutative semigroup rings. University of Chicago Press, Chicago–London, 1984

Chapter 163

Morita equivalence In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. It is named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958.

163.1 Motivation Rings are commonly studied in terms of their modules, as modules can be viewed as representations of rings. Every ring R has a natural R-module structure on itself where the module action is defined as the multiplication in the ring, so the approach via modules is more general and gives useful information. Because of this, one often studies a ring by studying the category of modules over that ring. Morita equivalence takes this viewpoint to a natural conclusion by defining rings to be Morita equivalent if their module categories are equivalent in the sense of category theory. This notion is of interest only when dealing with noncommutative rings, since it can be shown that two commutative rings are Morita equivalent if and only if they are isomorphic.

163.2 Definition Two rings R and S (associative, with 1) are said to be (Morita) equivalent if there is an equivalence of the category of (left) modules over R, R-Mod, and the category of (left) modules over S, S-Mod. It can be shown that the left module categories R-Mod and S-Mod are equivalent if and only if the right module categories Mod-R and Mod-S are equivalent. Further it can be shown that any functor from R-Mod to S-Mod that yields an equivalence is automatically additive.

163.3 Examples Any two isomorphic rings are Morita equivalent. The ring of n-by-n matrices with elements in R, denoted Mn(R), is Morita-equivalent to R for any n > 0. Notice that this generalizes the classification of simple artinian rings given by Artin–Wedderburn theory. To see the equivalence, notice that if X is a left R-module then Xn is an Mn(R)-module where the module structure is given by matrix multiplication on the left of column vectors from X. This allows the definition of a functor from the category of left R-modules to the category of left Mn(R)-modules. The inverse functor is defined by realizing that for any Mn(R)module there is a left R-module X such that the Mn(R)-module is obtained from X as described above. 494

163.4. CRITERIA FOR EQUIVALENCE

495

163.4 Criteria for equivalence Equivalences can be characterized as follows: if F:R-Mod → S-Mod and G:S-Mod → R-Mod are additive (covariant) functors, then F and G are an equivalence if and only if there is a balanced (S,R)-bimodule P such that SP and PR are finitely generated projective generators and there are natural isomorphisms of the functors F(−) ∼ = P ⊗R − , and of the functors G(−) ∼ = Hom(S P, −). Finitely generated projective generators are also sometimes called progenerators for their module category.[1] For every right-exact functor F from the category of left-R modules to the category of left-S modules that commutes with direct sums, a theorem of homological algebra shows that there is a (S,R)-bimodule E such that the functor F(−) is naturally isomorphic to the functor E ⊗R − . Since equivalences are by necessity exact and commute with direct sums, this implies that R and S are Morita equivalent if and only if there are bimodules RMS and SNR such that M ⊗S N ∼ = R as (R,R) bimodules and N ⊗R M ∼ = S as (S,S) bimodules. Moreover, N and M are related via an (S,R) bimodule isomorphism: N ∼ = Hom(MS , SS ) . ∼ End(PR ) for a progenerator module More concretely, two rings R and S are Morita equivalent if and only if S = PR,[2] which is the case if and only if S∼ = eMn (R)e (isomorphism of rings) for some positive integer n and full idempotent e in the matrix ring M (R). It is known that if R is Morita equivalent to S, then the ring C(R) is isomorphic to the ring C(S), where C(-) denotes the center of the ring, and furthermore R/J(R) is Morita equivalent to S/J(S), where J(-) denotes the Jacobson radical. While isomorphic rings are Morita equivalent, Morita equivalent rings can be nonisomorphic. An easy example is that a division ring D is Morita equivalent to all of its matrix rings Mn(D), but cannot be isomorphic when n > 1. In the special case of commutative rings, Morita equivalent rings are actually isomorphic. This follows immediately from the comment above, for if R is Morita equivalent to S, R = C(R) ∼ = C(S) = S . In fact, if R and S are isomorphic commutative rings, every equivalence between R-Mod and S-Mod arises up to natural isomorphism from an isomorphism between R and S.

163.5 Properties preserved by equivalence Many properties are preserved by the equivalence functor for the objects in the module category. Generally speaking, any property of modules defined purely in terms of modules and their homomorphisms (and not to their underlying elements or ring) is a categorical property which will be preserved by the equivalence functor. For example, if F(-) is the equivalence functor from R-Mod to S-Mod, then the R module M has any of the following properties if and only if the S module F(M) does: injective, projective, flat, faithful, simple, semisimple, finitely generated, finitely presented, Artinian, and Noetherian. Examples of properties not necessarily preserved include being free, and being cyclic. Many ring theoretic properties are stated in terms of their modules, and so these properties are preserved between Morita equivalent rings. Properties shared between equivalent rings are called Morita invariant properties. For example, a ring R is semisimple if and only if all of its modules are semisimple, and since semisimple modules are preserved under Morita equivalence, an equivalent ring S must also have all of its modules semisimple, and therefore be a semisimple ring itself. Sometimes it is not immediately obvious why a property should be preserved. For example, using one standard definition of von Neumann regular ring (for all a in R, there exists x in R such that a = axa) it is not clear that an equivalent ring should also be von Neumann regular. However another formulation is: a ring is von Neumann regular if and only if all of its modules are flat. Since flatness is preserved across Morita equivalence, it is now clear that von Neumann regularity is Morita invariant. The following properties are Morita invariant: • simple, semisimple • von Neumann regular • right (or left) Noetherian, right (or left) Artinian

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• right (or left) self-injective • quasi-Frobenius • prime, right (or left) primitive, semiprime, semiprimitive • right (or left) (semi-)hereditary • right (or left) nonsingular • right (or left) coherent • semiprimary, right (or left) perfect, semiperfect • semilocal Examples of properties which are not Morita invariant include commutative, local, reduced, domain, right (or left) Goldie, Frobenius, invariant basis number, and Dedekind finite. There are at least two other tests for determining whether or not a ring property P is Morita invariant. An element e in a ring R is a full idempotent when e2 = e and ReR = R. • P is Morita invariant if and only if whenever a ring R satisfies P , then so does eRe for every full idempotent e and so does every matrix ring M (R) for every positive integer n; or • P is Morita invariant if and only if: for any ring R and full idempotent e in R, R satisfies P if and only if the ring eRe satisfies P .

163.6 Further directions Dual to the theory of equivalences is the theory of dualities between the module categories, where the functors used are contravariant rather than covariant. This theory, though similar in form, has significant differences because there is no duality between the categories of modules for any rings, although dualities may exist for subcategories. In other words, because infinite dimensional modules are not generally reflexive, the theory of dualities applies more easily to finitely generated algebras over noetherian rings. Perhaps not surprisingly, the criterion above has an analogue for dualities, where the natural isomorphism is given in terms of the hom functor rather than the tensor functor. Morita Equivalence can also be defined in more structured situations, such as for symplectic groupoids and C*algebras. In the case of C*-algebras, a stronger type equivalence, called strong Morita equivalence, is needed to obtain results useful in applications, because of the additional structure of C*-algebras (coming from the involutive *-operation) and also because C*-algebras do not necessarily have an identity element.

163.7 Significance in K-theory If two rings are Morita equivalent, there is an induced equivalence of the respective categories of projective modules since the Morita equivalences will preserve exact sequences (and hence projective modules). Since the algebraic Ktheory of a ring is defined (in Quillen’s approach) in terms of the homotopy groups of (roughly) the classifying space of the nerve of the (small) category of finitely generated projective modules over the ring, Morita equivalent rings must have isomorphic K-groups.

163.8 References [1] DeMeyer & Ingraham (1971) p.6 [2] DeMeyer & Ingraham (1971) p.16

163.9. FURTHER READING

497

• Morita, Kiiti (1958). “Duality for modules and its applications to the theory of rings with minimum condition”. Science reports of the Tokyo Kyoiku Daigaku. Section A 6 (150): 83–142. ISSN 0371-3539. Zbl 0080.25702. • DeMeyer, F.; Ingraham, E. (1971). Separable algebras over commutative rings. Lecture Notes in Mathematics 181. Springer-Verlag. ISBN 978-3-540-05371-2. Zbl 0215.36602. • Anderson, F.W.; Fuller, K.R. (1992). Rings and Categories of Modules. Graduate Texts in Mathematics 13 (2nd ed.). New York: Springer-Verlag. ISBN 0-387-97845-3. Zbl 0765.16001. • Lam, T.Y. (2001). A first course in noncommutative rings. Graduate Texts in Mathematics 131 (2nd ed.). New York, NY: Springer-Verlag. Chapters 17-18-19. ISBN 0-387-95183-0. Zbl 0980.16001. • Meyer, Ralf. “Morita Equivalence In Algebra And Geometry”.

163.9 Further reading • Reiner, I. (2003). Maximal Orders. London Mathematical Society Monographs. New Series 28. Oxford University Press. pp. 154–169. ISBN 0-19-852673-3. Zbl 1024.16008.

Chapter 164

Multivector “p-vector” redirects here. For other uses, see K-vector (disambiguation). A multivector is the result of a product defined for elements in a vector space V. A vector space with a linear product operation between vectors is called an algebra; examples are matrix algebra and vector algebra.[1][2][3] The algebra of multivectors is constructed using the wedge product ∧ and is related to the exterior algebra of differential forms.[4] The set of multivectors on a vector space V is graded by the number of basis vectors that form a basis multivector. A multivector that is the product of p basis vectors is called a rank p multivector, or a p-vector. The linear combination of basis p-vectors forms a vector space denoted as Λp (V). The maximum rank of a multivector is the dimension of the vector space V. The product of a p-vector and a k-vector is a (k+p)-vector so the set of linear combinations of all multivectors on V is an associative algebra, which is closed with respect to the wedge product. This algebra, denoted by Λ(V), is called the exterior algebra of V.[5]

164.1 Wedge product The wedge product operation used to construct multivectors is linear, associative and alternating, which reflect the properties of the determinant. This means for vectors u, v and w in a vector space V and for scalars α, β, the wedge product has the properties, • Linear: u ∧ (αv + βw) = αu ∧ v + βu ∧ w; • Associative: (u ∧ v) ∧ w = u ∧ (v ∧ w) = u ∧ v ∧ w; • Alternating: u ∧ v = −v ∧ u,

u ∧ u = 0.

The product of p vectors is called a rank p multivector, or a p-vector. The maximum rank of a multivector is the dimension of the vector space V. The linearity of the wedge product allows a multivector to be defined as the linear combination of basis multivectors. There are (n p) basis p-vectors in an n-dimensional vector space.[4]

164.2 Area and volume The p-vector obtained from the wedge product of p separate vectors in an n-dimensional space has components that define the projected (p−1)-volumes of the p-parallelopiped spanned by the vectors. The square root of the sum of the squares of these components defines the volume of the p-parallelopiped.[4][6] 498

164.2. AREA AND VOLUME

499

The following examples show that a bivector in two dimensions measures the area of a parallelogram, and the magnitude of a bivector in three dimensions also measures the area of a parallelogram. Similarly, a three-vector in three dimensions measures the volume of a parallelepiped. It is easy to check that the magnitude of a three-vector in four dimensions measures the volume of the parallelepiped spanned by these vectors.

164.2.1

Multivectors in R2

Properties of multivectors can be seen by considering the two dimensional vector space V = R2 . Let the basis vectors be e1 and e2 , so u and v are given by u = u1 e1 + u2 e2 ,

v = v1 e1 + v2 e2 ,

and the multivector u ∧ v, also called a bivector, is computed to be u u ∧ v = 1 u2

v1 e ∧ e2 . v2 1

The vertical bars denote the determinant of the matrix, which is the area of the parallelogram spanned by the vectors u and v. The magnitude of u ∧ v is the area of this parallelogram. Notice that because V has dimension two the basis bivector e1 ∧ e2 is the only multivector in ΛV. The relationship between the magnitude of a multivector and the area or volume spanned by the vectors is an important feature in all dimensions. Furthermore, the linear functional version of a multivector that computes this volume is known as a differential form.

164.2.2

Multivectors in R3

More features of multivectors can be seen by considering the three dimensional vector space V = R3 . In this case, let the basis vectors be e1 , e2 , and e3 , so u, v and w are given by u = u1 e1 + u2 e2 + u3 e3 ,

v = v1 e1 + v2 e2 + v3 e3 ,

w = w1 e1 + w2 e2 + w3 e3 ,

and the bivector u ∧ v is computed to be u u ∧ v = 2 u3

u v2 e2 ∧ e3 + 1 v3 u3

u v1 e1 ∧ e3 + 1 v3 u2

v1 e ∧ e2 . v2 1

The components of this bivector are the same as the components of the cross product. The magnitude of this bivector is the square root of the sum of the squares of its components. This shows that the magnitude of the bivector u ∧ v is the area of the parallelogram spanned by the vectors u and v as it lies in the three-dimensional space V. The components of the bivector are the projected areas of the parallelogram on each of the three coordinate planes. Notice that because V has dimension three, there is one basis three-vector in ΛV. Compute the three-vector u1 u ∧ v ∧ w = u2 u3

v1 v2 v3

w1 w2 e1 ∧ e2 ∧ e3 . w3

This shows that the magnitude of the three-vector u ∧ v ∧ w is the volume of the parallelepiped spanned by the three vectors u, v and w. In higher-dimensional spaces, the component three-vectors are projections of the volume of a parallelepiped onto the coordinate three-spaces, and the magnitude of the three-vector is the volume of the parallelepiped as it sits in the higher-dimensional space.

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CHAPTER 164. MULTIVECTOR

164.3 Grassmann coordinates In this section, we consider multivectors on a projective space P n , which provide a convenient set of coordinates for lines, planes and hyperplanes that have properties similar to the homogeneous coordinates of points, called Grassmann coordinates.[7] Points in a real projective space P n are defined to be lines through the origin of the vector space Rn+1 . For example, the projective plane P 2 is the set of lines through the origin of R3 . Thus, multivectors defined on Rn+1 can be viewed as multivectors on P n . A convenient way to view a multivector on P n is to examine it in an affine component of P n , which is the intersection of the lines through the origin of Rn+1 with a selected hyperplane, such as H: xn₊₁ = 1. Lines through the origin of R3 intersect the plane E: z = 1 to define an affine version of the projective plane that only lacks the points z = 0, called the points at infinity.

164.3.1

Multivectors on P 2

Points in the affine component E: z = 1 of the projective plane have coordinates x = (x, y, 1). A linear combination of two points p = (p1 , p1 , 1) and q = (q1 , q1 , 1) defines a plane in R3 that intersects E in the line joining p and q. The multivector p ∧ q defines a parallelogram in R3 given by

p ∧ q = (p2 − q2 )e2 ∧ e3 + (p1 − q1 )e1 ∧ e3 + (p1 q2 − q1 p2 )e1 ∧ e2 . Notice that substitution of αp + βq for p multiplies this multivector by a constant. Therefore, the components of p ∧ q are homogeneous coordinates for the plane through the origin of R3 . The set of points x = (x, y, 1) on the line through p and q is the intersection of the plane defined by p ∧ q with the plane E: z = 1. These points satisfy x ∧ p ∧ q = 0, that is, ( ) x ∧ p ∧ q = (xe1 + ye2 + e3 ) ∧ (p2 − q2 )e2 ∧ e3 + (p1 − q1 )e1 ∧ e3 + (p1 q2 − q1 p2 )e1 ∧ e2 = 0, which simplifies to the equation of a line

λ : x(p2 − q2 ) + y(p1 − q1 ) + (p1 q2 − q1 p2 ) = 0. This equation is satisfied by points x = αp + βq for real values of α and β. The three components of p ∧ q that define the line λ are called the Grassmann coordinates of the line. Because three homogeneous coordinates define both a point and a line, the geometry of points is said to be dual to the geometry of lines in the projective plane. This is called the principle of duality.

164.3.2

Multivectors on P 3

Three dimensional projective space, P 3 consists of all lines through the origin of R4 . Let the three dimensional hyperplane, H: w = 1, be the affine component of projective space defined by the points x = (x, y, z, 1). The multivector p ∧ q ∧ r defines a parallelepiped in R4 given by p2 p∧q∧r = p3 1

q2 q3 1

p1 r2 r3 e2 ∧e3 ∧e4 + p3 1 1

q1 q3 1

p1 r1 r3 e1 ∧e3 ∧e4 + p2 1 1

q1 q2 1

p1 r1 r2 e1 ∧e2 ∧e4 + p2 p3 1

q1 q2 q3

r1 r2 e1 ∧e2 ∧e3 . r3

Notice that substitution of αp + βq + γr for p multiplies this multivector by a constant. Therefore, the components of p ∧ q ∧ r are homogeneous coordinates for the 3-space through the origin of R4 . A plane in the affine component H: w = 1 is the set of points x = (x, y, z, 1) in the intersection of H with the 3-space defined by p ∧ q ∧ r. These points satisfy x ∧ p ∧ q ∧ r = 0, that is,

164.4. CLIFFORD PRODUCT

501

x ∧ p ∧ q ∧ r = (xe1 + ye2 + ze3 + e4 ) ∧ p ∧ q ∧ r = 0, which simplifies to the equation of a plane p2 λ : x p3 1

q2 q3 1

p1 r2 r3 + y p3 1 1

q1 q3 1

p1 r1 r3 + z p2 1 1

q1 q2 1

r1 p1 r2 + p2 1 p3

q1 q2 q3

r1 r2 = 0. r3

This equation is satisfied by points x = αp + βq + γr for real values of α, β and γ. The four components of p ∧ q ∧ r that define the plane λ are called the Grassmann coordinates of the plane. Because four homogeneous coordinates define both a point and a plane in projective space, the geometry of points is dual to the geometry of planes. A line as the join of two points: In projective space the line λ through two points p and q can be viewed as the intersection of the affine space H: w = 1 with the plane x = αp + βq in R4 . The multivector p ∧ q provides homogeneous coordinates for the line

λ : p ∧ q = (p1 e1 + p2 e2 + p3 e3 + e4 ) ∧ (q1 e1 + q2 e2 + q3 e3 + e4 ), p1 q1 p2 q2 p3 q3 p2 q2 p = e1 ∧ e4 + e2 ∧ e4 + e3 ∧ e4 + e2 ∧ e3 + 3 1 1 1 1 1 1 p 3 q3 p1

p q3 e3 ∧ e1 + 1 q1 p2

q1 e ∧e . q2 1 2

These are known as the Plücker coordinates of the line, though they are also an example of Grassmann coordinates. A line as the intersection of two planes: A line μ in projective space can also be defined as the set of points x that form the intersection of two planes π and ρ defined by rank three multivectors, so the points x are the solutions to the linear equations

µ : x ∧ π = 0, x ∧ ρ = 0. In order to obtain the Plucker coordinates of the line μ, map the multivectors π and ρ to their dual point coordinates using the Hodge star operator,[4]

e1 = ∗e2 ∧ e3 ∧ e4 , −e2 = ∗e1 ∧ e3 ∧ e4 , e3 = ∗e1 ∧ e2 ∧ e4 , −e4 = ∗e1 ∧ e2 ∧ e3 , then

∗π = π1 e1 + π2 e2 + π3 e3 + π4 e4 ,

∗ρ = ρ1 e1 + ρ2 e2 + ρ3 e3 + ρ4 e4 .

So, the Plücker coordinates of the line μ are given by π µ : (∗π)∧(∗ρ) = 1 π4

π ρ1 e ∧e + 2 ρ4 1 4 π4

π ρ2 e ∧e + 3 ρ4 2 4 π4

π ρ3 e ∧e + 2 ρ4 3 4 π3

π ρ2 e ∧e + 3 ρ3 2 3 π1

π ρ3 e ∧e + 1 ρ1 3 1 π2

ρ1 e ∧e . ρ2 1 2

Because the six homogeneous coordinates of a line can be obtained from the join of two points or the intersection of two planes, the line is said to be self dual in projective space.

164.4 Clifford product W. K. Clifford combined multivectors with the inner product defined on the vector space, in order to obtain a general construction for hypercomplex numbers that includes the usual complex numbers and Hamilton’s quaternions.[8][9]

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CHAPTER 164. MULTIVECTOR

The Clifford product between two vectors u and v is linear and associative like the wedge product, and has the additional property that the multivector uv is coupled to the inner product u · v by Clifford’s relation,

uv + vu = −2u · v. Clifford’s relation preserves the alternating property for the product of vectors that are perpendicular. This can be seen for the orthogonal unit vectors ei, i = 1, ..., n in Rn . Clifford’s relation yields

ei ej + ej ei = −2ei · ej = 0, therefore the basis vectors are alternating,

ei ej = −ej ei ,

i ̸= j = 1, . . . , n.

In contrast to the wedge product, the Clifford product of a vector with itself is no longer zero. To see this compute the product,

ei ei + ei ei = −2ei · ei = −2, which yields

ei ei = −1,

i = 1, . . . , n.

The set of multivectors constructed using Clifford’s product yields an associative algebra known as a Clifford algebra. Inner products with different properties can be used to construct different Clifford algebras.[10][11]

164.5 Geometric algebra See also: Blade (geometry) Multivectors play a central role in the mathematical formulation of physics known as geometric algebra. The term geometric algebra was used by E. Artin for matrix methods in projective geometry.[12] It was D. Hestenes who used geometric algebra to describe the application of Clifford algebras to classical mechanics,[13] This formulation was expanded to geometric calculus by D. Hestenes and G. Sobczyk,[14] who provided new terminology for a variety of features in this application of Clifford algebra to physics. C. Doran and A. Lasenby show that Hestene’s geometric algebra provides a convenient formulation for modern physics.[15] In geometric algebra, a multivector is defined to be the sum of different-grade k-blades, such as the summation of a scalar, a vector, and a 2-vector.[16] A sum of only k-grade components is called a k-vector,[17] or a homogeneous multivector.[18] The highest grade element in a space is called a pseudoscalar. If a given element is homogeneous of a grade k, then it is a k-vector, but not necessarily a k-blade. Such an element is a k-blade when it can be expressed as the wedge product of k vectors. A geometric algebra generated by a 4dimensional Euclidean vector space illustrates the point with an example: The sum of any two blades with one taken from the XY-plane and the other taken from the ZW-plane will form a 2-vector that is not a 2-blade. In a geometric algebra generated by a Euclidean vector space of dimension 2 or 3, all sums of 2-blades may be written as a single 2-blade.

164.5. GEOMETRIC ALGEBRA

164.5.1

503

Examples

Orientation defined by an ordered set of vectors.

Reversed orientation corresponds to negating the exterior product. Geometric interpretation of grade n elements in a real exterior algebra for n = 0 (signed point), 1 (directed line segment, or vector), 2 (oriented plane element), 3 (oriented volume). The exterior product of n vectors can be visualized as any n-dimensional shape (e.g. n-parallelotope, n-ellipsoid); with magnitude (hypervolume), and orientation defined by that on its n − 1-dimensional boundary and on which side the interior is.[19][20]

• 0-vectors are scalars; • 1-vectors are vectors; • 2-vectors are bivectors; • (n − 1)-vectors are pseudovectors; • n-vectors are pseudoscalars. In the presence of a volume form (such as given an inner product and an orientation), pseudovectors and pseudoscalars can be identified with vectors and scalars, which is routine in vector calculus, but without a volume form this cannot be done without a choice. In the Algebra of physical space (the geometric algebra of Euclidean 3-space, used as a model of (3+1)-spacetime), a sum of a scalar and a vector is called a paravector, and represents a point in spacetime (the vector the space, the scalar the time).

504

CHAPTER 164. MULTIVECTOR

164.5.2

Bivectors

Main article: Bivector A bivector is therefore an element of the antisymmetric tensor product of a tangent space with itself. In geometric algebra, also, a bivector is a grade 2 element (a 2-vector) resulting from the wedge product of two vectors, and so it is geometrically an oriented area, in the same way a vector is an oriented line segment. If a and b are two vectors, the bivector a ∧ b has • a norm which is its area, given by

∥a ∧ b∥ = ∥a∥ ∥b∥ sin(ϕa,b ) • a direction: the plane where that area lies on, i.e., the plane determined by a and b, as long as they are linearly independent; • an orientation (out of two), determined by the order in which the originating vectors are multiplied. Bivectors are connected to pseudovectors, and are used to represent rotations in geometric algebra. As bivectors are elements of a vector space Λ2 V (where V is a finite-dimensional vector space with {{{1}}}), it makes sense to define an inner product on this vector space as follows. First, write any element F ∈ Λ2 V in terms of a basis (ei ∧ ej)₁ ≤ i < j ≤ n of Λ2 V as

F = F ab ea ∧ eb

(1 ≤ a < b ≤ n)

where the Einstein summation convention is being used. Now define a map G : Λ2 V × Λ2 V → R by insisting that

G(F, H) := Gabcd F ab H cd where Gabcd are a set of numbers.

164.6 Applications Bivectors play many important roles in physics, for example, in the classification of electromagnetic fields.

164.7 See also • Blade (geometry) • Paravector

164.8 References [1] F. E. Hohn, Elementary Matrix Algebra, Dover Publications, 2011 [2] H. Kishan, Vector Algebra and Calculus, Atlantic Publ., 2007 [3] L. Brand, Vector Analysis, Dover Publications, 2006

164.8. REFERENCES

505

[4] H. Flanders, Differential Forms with Applications to the Physical Sciences, Academic Press, New York, NY, 1963 [5] Baylis (1994). Theoretical methods in the physical sciences: an introduction to problem solving using Maple V. Birkhäuser. p. 234, see footnote. ISBN 0-8176-3715-X. [6] G. E. Shilov, Linear Algebra, (trans. R. A. Silverman), Dover Publications, 1977. [7] W. V. D. Hodge and D. Pedoe, Methods of Algebraic Geometry, Vol. 1, Cambridge Univ. Press, 1947 [8] W. K. Clifford, “Preliminary sketch of bi-quaternions,” Proc. London Math. Soc. Vol. 4 (1873) pp. 381-395 [9] W. K. Clifford, Mathematical Papers, (ed. R. Tucker), London: Macmillan, 1882. [10] J. M. McCarthy, An Introduction to Theoretical Kinematics, pp. 62–5, MIT Press 1990. [11] O. Bottema and B. Roth, Theoretical Kinematics, North Holland Publ. Co., 1979 [12] E. Artin, Geometric algebra, Interscience Publ., 1957 [13] D. Hestenes, New Foundations for Classical Mechanics, Kluwer Academic Publishers, 1986. [14] D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, Springer Verlag, 1987 [15] C. Doran and A. Lasenby, Geometric Algebra for Physicists, Cambridge Univ. Press, 2007. [16] Marcos A. Rodrigues (2000). "§1.2 Geometric algebra: an outline”. Invariants for pattern recognition and classification. World Scientific. p. 3 ff. ISBN 981-02-4278-6. [17] R Wareham, J Cameron & J Lasenby (2005). “Applications of conformal geometric algebra in computer vision and graphics”. In Hongbo Li, Peter J. Olver, Gerald Sommer. Computer algebra and geometric algebra with applications. Springer. p. 330. ISBN 3-540-26296-2. [18] Eduardo Bayro-Corrochano (2004). “Clifford geometric algebra: A promising framework for computer vision, robotics and learning”. In Alberto Sanfeliu, José Francisco Martínez Trinidad, Jesús Ariel Carrasco Ochoa. Progress in pattern recognition, image analysis and applications. Springer. p. 25. ISBN 3-540-23527-2. [19] R. Penrose (2007). The Road to Reality. Vintage books. ISBN 0-679-77631-1. [20] J.A. Wheeler, C. Misner, K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. 83. ISBN 0-7167-0344-0.

Chapter 165

Nakayama algebra In algebra, a Nakayama algebra or generalized uniserial algebra is an algebra such that the left and right projective modules have a unique composition series (Reiten 1982, p. 39). They were studied by Tadasi Nakayama (1940) who called them “generalized uni-serial rings”. An example of a Nakayama algebra is k[x]/(xn ) for k a field and n a positive integer. Current usage of uniserial differs slightly: an explanation of the difference appears here.

165.1 References • Nakayama, Tadasi (1940), “Note on uni-serial and generalized uni-serial rings”, Proc. Imp. Acad. Tokyo 16: 285–289, MR 0003618 • Reiten, Idun (1982), “The use of almost split sequences in the representation theory of Artin algebras”, Representations of algebras (Puebla, 1980), Lecture Notes in Math. 944, Berlin, New York: Springer-Verlag, pp. 29–104, doi:10.1007/BFb0094057, MR 672115

506

Chapter 166

Nakayama’s conjecture In mathematics, Nakayama’s conjecture is a conjecture about Artinian rings, introduced by Nakayama (1958). The generalized Nakayama conjecture is an extension to more general rings, introduced by Auslander and Reiten (1975). Leuschke & Huneke (2004) proved some cases of the generalized Nakayama conjecture. Nakayama’s conjecture states that if all the modules of a minimal injective resolution of an Artin algebra R are injective and projective, then R is self-injective.

166.1 References • Auslander, Maurice; Reiten, Idun (1975), “On a generalized version of the Nakayama conjecture”, Proceedings of the American Mathematical Society 52: 69–74, doi:10.2307/2040102, ISSN 0002-9939, MR 0389977 • Leuschke, Graham J.; Huneke, Craig (2004), “On a conjecture of Auslander and Reiten”, Journal of Algebra 275 (2): 781–790, doi:10.1016/j.jalgebra.2003.07.018, ISSN 0021-8693, MR 2052636 • Nakayama, Tadasi (1958), “On algebras with complete homology”, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 22: 300–307, doi:10.1007/BF02941960, ISSN 0025-5858, MR 0104718

507

Chapter 167

Necklace ring In mathematics, the necklace ring is a ring introduced by Metropolis and Rota (1983).

167.1 Definition If A is a commutative ring then the necklace ring over A consists of all infinite sequences (a1 ,a2 ,...) of elements of A. Addition in the necklace ring is given by pointwise addition of sequences. Multiplication is given by a sort of arithmetic convolution: the product of (a1 ,a2 ,...) and (b1 ,b2 ,...) has components

cn =

∑

(i, j)ai bj

[i,j]=n

where [i,j] is the least common multiple of i and j, and (i,j) is their highest common factor.

167.2 See also • Necklace polynomial

167.3 References • Hazewinkel, Michiel (2009). “Witt vectors. I.”. Handbook of Algebra 6. Elsevier/North-Holland. pp. 319– 472. arXiv:0804.3888. ISBN 978-0-444-53257-2. MR 2553661. • Metropolis, N.; Rota, Gian-Carlo (1983). “Witt vectors and the algebra of necklaces”. Advances in Mathematics 50 (2): 95–125. doi:10.1016/0001-8708(83)90035-X. MR 723197.

508

Chapter 168

Nil ideal In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if each of its elements is nilpotent.[1][2] The nilradical of a commutative ring is an example of a nil ideal; in fact, it is the ideal of the ring maximal with respect to the property of being nil. Unfortunately the set of nil elements does not always form an ideal for noncommutative rings. Nil ideals are still associated with interesting open questions, especially the unsolved Köthe conjecture.

168.1 Commutative rings In a commutative ring, the set of all nilpotent elements forms an ideal known as the nilradical of the ring. Therefore, an ideal of a commutative ring is nil if, and only if, it is a subset of the nilradical; that is, the nilradical is the ideal maximal with respect to the property that each of its elements is nilpotent. In commutative rings, the nil ideals are more well-understood compared to the case of noncommutative rings. This is primarily because the commutativity assumption ensures that the product of two nilpotent elements is again nilpotent. For instance, if a is a nilpotent element of a commutative ring R, a·R is an ideal that is in fact nil. This is because any element of the principal ideal generated by a is of the form a·r for r in R, and if an = 0, (a·r)n = an ·rn = 0. It is not in general true however, that a·R is a nil (one-sided) ideal in a noncommutative ring, even if a is nilpotent.

168.2 Noncommutative rings The theory of nil ideals is of major importance in noncommutative ring theory. In particular, through the understanding of nil rings—rings whose every element is nilpotent—one may obtain a much better understanding of more general rings.[3] In the case of commutative rings, there is always a maximal nil ideal: the nilradical of the ring. The existence of such a maximal nil ideal in the case of noncommutative rings is guaranteed by the fact that the sum of nil ideals is again nil. However, the truth of the assertion that the sum of two left nil ideals is again a left nil ideal remains elusive; it is an open problem known as the Köthe conjecture.[4] The Köthe conjecture was first posed in 1930 and yet remains unresolved as of 2010.

168.3 Relation to nilpotent ideals The notion of a nil ideal has a deep connection with that of a nilpotent ideal, and in some classes of rings, the two notions coincide. If an ideal is nilpotent, it is of course nil. There are two main barriers for nil ideals to be nilpotent: 1. There need not be an upper bound on the exponent required to annihilate elements. Arbitrarily high exponents may be required. 2. The product of n nilpotent elements may be nonzero for arbitrarily high n. 509

510

CHAPTER 168. NIL IDEAL

Clearly both of these barriers must be avoided for a nil ideal to qualify as nilpotent. In a right artinian ring, any nil ideal is nilpotent.[5] This is proven by observing that any nil ideal is contained in the Jacobson radical of the ring, and since the Jacobson radical is a nilpotent ideal (due to the artinian hypothesis), the result follows. In fact, this has been generalized to right noetherian rings; the result is known as Levitzky’s theorem. A particularly simple proof due to Utumi can be found in (Herstein 1968, Theorem 1.4.5, p. 37).

168.4 See also • Köthe conjecture • Nilpotent ideal • Nilradical • Jacobson radical

168.5 Notes [1] Isaacs 1993, p. 194 [2] Herstein 1968, Definition (b), p. 13 [3] Section 2 of Smoktunowicz 2006, p. 260 [4] Herstein 1968, p. 21 [5] Isaacs, Corollary 14.3, p. 195.

168.6 References • Herstein, I. N. (1968), Noncommutative rings (1st ed.), The Mathematical Association of America, ISBN 088385-015-X • Isaacs, I. Martin (1993), Algebra, a graduate course (1st ed.), Brooks/Cole Publishing Company, ISBN 0-53419002-2 • Smoktunowicz, Agata (2006), “Some results in noncommutative ring theory” (PDF), International Congress of Mathematicians, Vol. II, Zürich: European Mathematical Society, pp. 259–269, ISBN 978-3-03719-022-7, MR 2275597, retrieved 2009-08-19

Chapter 169

Nilpotent This article is about a type of element in a ring. For the type of group, see Nilpotent group. For the type of ideal, see Nilpotent ideal. In mathematics, an element, x, of a ring, R, is called nilpotent if there exists some positive integer, n, such that xn = 0. The term was introduced by Benjamin Peirce in the context of his work on the classification of algebras.[1]

169.1 Examples • This definition can be applied in particular to square matrices. The matrix



0 1 A = 0 0 0 0

 0 1 0

is nilpotent because A3 = 0. See nilpotent matrix for more. • In the factor ring Z/9Z, the equivalence class of 3 is nilpotent because 32 is congruent to 0 modulo 9. • Assume that two elements a, b in a (non-commutative) ring R satisfy ab = 0. Then the element c = ba is nilpotent (if non-zero) as c2 = (ba)2 = b(ab)a = 0. An example with matrices (for a, b):

( A=

) ( 0 1 0 , B= 0 1 0

) 1 . 0

Here AB = 0, BA = B. • The ring of coquaternions contains a cone of nilpotents.

169.2 Properties No nilpotent element can be a unit (except in the trivial ring {0} which has only a single element 0 = 1). All non-zero nilpotent elements are zero divisors. An n-by-n matrix A with entries from a field is nilpotent if and only if its characteristic polynomial is t n . 511

512

CHAPTER 169. NILPOTENT

If x is nilpotent, then 1 − x is a unit, because xn = 0 entails (1 − x)(1 + x + x2 + · · · + xn−1 ) = 1 − xn = 1. More generally, the sum of a unit element and a nilpotent element is a unit when they commute.

169.3 Commutative rings The nilpotent elements from a commutative ring R form an ideal N ; this is a consequence of the binomial theorem. This ideal is the nilradical of the ring. Every nilpotent element x in a commutative ring is contained in every prime ideal p of that ring, since xn = 0 ∈ p . So N is contained in the intersection of all prime ideals. If x is not nilpotent, we are able to localize with respect to the powers of x : S = {1, x, x2 , ...} to get a non-zero ring S −1 R . The prime ideals of the localized ring correspond exactly to those primes p with p ∩ S = ∅ .[2] As every non-zero commutative ring has a maximal ideal, which is prime, every non-nilpotent x is not contained in some prime ideal. Thus N is exactly the intersection of all prime ideals.[3] A characteristic similar to that of Jacobson radical and annihilation of simple modules is available for nilradical: nilpotent elements of ring R are precisely those that annihilate all integral domains internal to the ring R (that is, of the form R/I for prime ideals I). This follows from the fact that nilradical is the intersection of all prime ideals.

169.4 Nilpotent elements in Lie algebra Let g be a Lie algebra. Then an element of g is called nilpotent if it is in [g, g] and ad x is a nilpotent transformation. See also: Jordan decomposition in a Lie algebra.

169.5 Nilpotency in physics An operand Q that satisfies Q2 = 0 is nilpotent. Grassmann numbers which allow a path integral representation for Fermionic fields are nilpotents since their squares vanish. The BRST charge is an important example in physics. As linear operators form an associative algebra and thus a ring, this is a special case of the initial definition.[4][5] More generally, in view of the above definitions, an operator Q is nilpotent if there is n∈N such that Qn = 0 (the zero function). Thus, a linear map is nilpotent iff it has a nilpotent matrix in some basis. Another example for this is the exterior derivative (again with n = 2). Both are linked, also through supersymmetry and Morse theory,[6] as shown by Edward Witten in a celebrated article.[7] The electromagnetic field of a plane wave without sources is nilpotent when it is expressed in terms of the algebra of physical space.[8]

169.6 Algebraic nilpotents The two-dimensional dual numbers contain a nilpotent space. Other algebras and numbers that contain nilpotent spaces include split-quaternions (coquaternions), split-octonions, biquaternions C⊗H , and complex octonions C⊗O .

169.7 See also • Idempotent element • Unipotent • Reduced ring • Nil ideal

169.8. REFERENCES

513

169.8 References [1] Polcino Milies & Sehgal (2002), An Introduction to Group Rings. p. 127. [2] Matsumura, Hideyuki (1970). “Chapter 1: Elementary Results”. Commutative Algebra. W. A. Benjamin. p. 6. ISBN 978-0-805-37025-6. [3] Atiyah, M. F.; MacDonald, I. G. (February 21, 1994). “Chapter 1: Rings and Ideals”. Introduction to Commutative Algebra. Westview Press. p. 5. ISBN 978-0-201-40751-8. [4] Peirce, B. Linear Associative Algebra. 1870. [5] Polcino Milies, César; Sehgal, Sudarshan K. An introduction to group rings. Algebras and applications, Volume 1. Springer, 2002. ISBN 978-1-4020-0238-0 [6] A. Rogers, The topological particle and Morse theory, Class. Quantum Grav. 17:3703–3714,2000 doi:10.1088/02649381/17/18/309. [7] E Witten, Supersymmetry and Morse theory. J.Diff.Geom.17:661–692,1982. [8] Rowlands, P. Zero to Infinity: The Foundations of Physics, London, World Scientific 2007, ISBN 978-981-270-914-1

Chapter 170

Nilpotent algebra (ring theory) In mathematics, specifically in ring theory, a nilpotent algebra over a commutative ring is an algebra over a commutative ring, in which for some positive integer n every product containing at least n elements of the algebra is zero. The concept of a nilpotent Lie algebra has a different definition, which depends upon the Lie bracket. (There is no Lie bracket for many algebras over commutative rings; a Lie algebra involves its Lie bracket, whereas, there is no Lie bracket defined in the general case of an algebra over a commutative ring.) Another possible source of confusion in terminology is the quantum nilpotent algebra,[1] a concept related to quantum groups and Hopf algebras.

170.1 Formal definition An associative algebra A over a commutative ring R is defined to be a nilpotent algebra if and only if there exists some positive integer n such that 0 = y1 y2 · · · yn for all y1 , y2 , . . . , yn in the algebra A . The smallest such n is called the index of the algebra A .[2] In the case of a non-associative algebra, the definition is that every different multiplicative association of the n elements is zero.

170.2 Nil algebra An algebra in which every element of the algebra is nilpotent is called a nil algebra.[3]

170.3 See also • Algebraic structure (a much more general term) • nil-Coxeter algebra • Lie algebra • Example of a non-associative algebra

170.4 References [1] Goodearl, K. R.; Yakimov, M. T. (1 Nov 2013). “Unipotent and Nakayama automorphisms of quantum nilpotent algebras”. arxiv.org. [2] Albert, A. Adrian (2003) [1939]. “Chapt. 2: Ideals and Nilpotent Algebras”. Structure of Algebras. Colloquium Publications, Col. 24. Amer. Math. Soc. p. 22. ISBN 0-8218-1024-3. ISSN 0065-9258; reprint with corrections of revised 1961 edition [3] Nil algebra – Encyclopedia of Mathematics

514

170.5. EXTERNAL LINKS

515

• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: SpringerVerlag, ISBN 978-0-387-95385-4, MR 1878556

170.5 External links • Nilpotent algebra – Encyclopedia of Mathematics

Chapter 171

Nilpotent ideal In mathematics, more specifically ring theory, an ideal, I, of a ring is said to be a nilpotent ideal, if there exists a natural number k such that I k = 0.[1] By I k , it is meant the additive subgroup generated by the set of all products of k elements in I.[1] Therefore, I is nilpotent if and only if there is a natural number k such that the product of any k elements of I is 0. The notion of a nilpotent ideal is much stronger than that of a nil ideal in many classes of rings. There are, however, instances when the two notions coincide—this is exemplified by Levitzky’s theorem.[2][3] The notion of a nilpotent ideal, although interesting in the case of commutative rings, is most interesting in the case of noncommutative rings.

171.1 Relation to nil ideals The notion of a nil ideal has a deep connection with that of a nilpotent ideal, and in some classes of rings, the two notions coincide. If an ideal is nilpotent, it is of course nil, but a nil ideal need not be nilpotent for more than one reason. The first is that there need not be a global upper bound on the exponent required to annihilate various elements of the nil ideal, and secondly, each element being nilpotent does not force products of distinct elements to vanish.[1] In a right artinian ring, any nil ideal is nilpotent.[4] This is proven by observing that any nil ideal is contained in the Jacobson radical of the ring, and since the Jacobson radical is a nilpotent ideal (due to the artinian hypothesis), the result follows. In fact, this can be generalized to right noetherian rings; this result is known as Levitzky’s theorem.[3]

171.2 See also • Köthe conjecture • Nilpotent element • Nil ideal • Nilradical • Jacobson radical

171.3 Notes [1] Isaacs, p. 194. [2] Isaacs, Theorem 14.38, p. 210 [3] Herstein, Theorem 1.4.5, p. 37. [4] Isaacs, Corollary 14.3, p. 195

516

171.4. REFERENCES

517

171.4 References • I.N. Herstein (1968). Noncommutative rings (1st edition ed.). The Mathematical Association of America. ISBN 0-88385-015-X. • I. Martin Isaacs (1993). Algebra, a graduate course (1st edition ed.). Brooks/Cole Publishing Company. ISBN 0-534-19002-2.

Chapter 172

Nilradical of a ring In algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements of the ring. In the non-commutative ring case the same definition does not always work. This has resulted in several radicals generalizing the commutative case in distinct ways. See the article "radical of a ring" for more of this. The nilradical of a Lie algebra is similarly defined for Lie algebras.

172.1 Commutative rings The nilradical of a commutative ring is the set of all nilpotent elements in the ring, or equivalently the radical of the zero ideal. This is an ideal because the sum of any two nilpotent elements is nilpotent (by the binomial formula), and the product of any element with a nilpotent element (by commutativity) is nilpotent. It can also be characterized as the intersection of all the prime ideals of the ring. (In fact, it is the intersection of all minimal prime ideals.) A ring is called reduced if it has no nonzero nilpotent. Thus, a ring is reduced if and only if its nilradical is zero. If R is an arbitrary commutative ring, then the quotient of it by the nilradical is a reduced ring and is denoted by Rred . Since every maximal ideal is a prime ideal, the Jacobson radical — which is the intersection of maximal ideals — must contain the nilradical. A ring is called a Jacobson ring if the nilradical of R/P coincides with the Jacobson radical of R/P for every prime ideal P of R. An Artinian ring is Jacobson, and its nilradical is the maximal nilpotent ideal of the ring. In general, if the nilradical is finitely generated (e.g., the ring is Noetherian), then it is nilpotent.

172.2 Noncommutative rings Further information: Radical of a ring For noncommutative rings, there are several analogues of the nilradical. The lower nilradical (or Baer–McCoy radical, or prime radical) is the analogue of the radical of the zero ideal and is defined as the intersection of the prime ideals of the ring. The analogue of the set of all nilpotent elements is the upper nilradical and is defined as the ideal generated by all nil ideals of the ring, which is itself a nil ideal. The set of all nilpotent elements itself need not be an ideal (or even a subgroup), so the upper nilradical can be much smaller than this set. The Levitzki radical is in between and is defined as the largest locally nilpotent ideal. As in the commutative case, when the ring is artinian, the Levitzki radical is nilpotent and so is the unique largest nilpotent ideal. Indeed, if the ring is merely noetherian, then the lower, upper, and Levitzki radical are nilpotent and coincide, allowing the nilradical of any noetherian ring to be defined as the unique largest (left, right, or two-sided) nilpotent ideal of the ring.

172.3 References • Eisenbud, David, “Commutative Algebra with a View Toward Algebraic Geometry”, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8. 518

172.3. REFERENCES

519

• Lam, Tsit-Yuen (2001), A First Course in Noncommutative Rings (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-95325-0, MR 1838439

Chapter 173

Noetherian module In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion. Historically, Hilbert was the first mathematician to work with the properties of finitely generated submodules. He proved an important theorem known as Hilbert’s basis theorem which says that any ideal in the multivariate polynomial ring of an arbitrary field is finitely generated. However, the property is named after Emmy Noether who was the first one to discover the true importance of the property.

173.1 Characterizations, properties and examples In the presence of the axiom of choice, two other characterizations are possible: • Any nonempty set S of submodules of the module has a maximal element (with respect to set inclusion.) This is known as the maximum condition. • All of the submodules of the module are finitely generated. If M is a module and K a submodule, then M is Noetherian if and only if K and M/K are Noetherian. This is in contrast to the general situation with finitely generated modules: a submodule of a finitely generated module need not be finitely generated. Examples • The integers, considered as a module over the ring of integers, is a Noetherian module. • If R = Mn(F) is the full matrix ring over a field, and M = Mn ₁(F) is the set of column vectors over F, then M can be made into a module using matrix multiplication by elements of R on the left of elements of M. This is a Noetherian module. • Any module that is finite as a set is Noetherian. • Any finitely generated right module over a right Noetherian ring is a Noetherian module.

173.2 Use in other structures A right Noetherian ring R is, by definition, a Noetherian right R module over itself using multiplication on the right. Likewise a ring is called left Noetherian ring when R is Noetherian considered as a left R module. When R is a commutative ring the left-right adjectives may be dropped, as they are unnecessary. Also, if R is Noetherian on both sides, it is customary to call it Noetherian and not “left and right Noetherian”. The Noetherian condition can also be defined on bimodule structures as well: a Noetherian bimodule is a bimodule whose poset of sub-bimodules satisfies the ascending chain condition. Since a sub-bimodule of an R-S bimodule 520

173.3. SEE ALSO

521

M is in particular a left R-module, if M considered as a left R module were Noetherian, then M is automatically a Noetherian bimodule. It may happen, however, that a bimodule is Noetherian without its left or right structures being Noetherian.

173.3 See also • Artinian module • Ascending/descending chain condition • Composition series • finitely generated module • Krull dimension

173.4 References • Eisenbud Commutative Algebra with a View Toward Algebraic Geometry, Springer-Verlag, 1995.

Chapter 174

Noetherian ring In mathematics, more specifically in the area of abstract algebra known as ring theory, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals; that is, given any chain: I1 ⊆ · · · ⊆ Ik−1 ⊆ Ik ⊆ Ik+1 ⊆ · · · there exists an n such that: In = In+1 = · · · . There are other equivalent formulations of the definition of a Noetherian ring and these are outlined later in the article. Noetherian rings are named after Emmy Noether. The notion of a Noetherian ring is of fundamental importance in both commutative and noncommutative ring theory, due to the role it plays in simplifying the ideal structure of a ring. For instance, the ring of integers and the polynomial ring over a field are both Noetherian rings, and consequently, such theorems as the Lasker–Noether theorem, the Krull intersection theorem, and the Hilbert’s basis theorem hold for them. Furthermore, if a ring is Noetherian, then it satisfies the descending chain condition on prime ideals. This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the Krull dimension.

174.1 Characterizations For noncommutative rings, it is necessary to distinguish between three very similar concepts: • A ring is left-Noetherian if it satisfies the ascending chain condition on left ideals. • A ring is right-Noetherian if it satisfies the ascending chain condition on right ideals. • A ring is Noetherian if it is both left- and right-Noetherian. For commutative rings, all three concepts coincide, but in general they are different. There are rings that are leftNoetherian and not right-Noetherian, and vice versa. There are other, equivalent, definitions for a ring R to be left-Noetherian: • Every left ideal I in R is finitely generated, i.e. there exist elements a1 , ..., an in I such that I = Ra1 + ... + Ran.[1] • Every non-empty set of left ideals of R, partially ordered by inclusion, has a maximal element with respect to set inclusion.[1] Similar results hold for right-Noetherian rings. For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated. (The result is due to I. S. Cohen.) 522

174.2. PROPERTIES

523

174.2 Properties • ℤ is a Noetherian ring, a fact which is exploited in the usual proof that every non-unit integer is divisible by at least one prime, although it’s usually stated as “every non-empty set of integers has a minimal element with respect to divisibility”. • If R is a Noetherian ring, then R[X] is Noetherian by the Hilbert basis theorem. By induction, R[X1 , ..., Xn] is a Noetherian ring. Also, R[[X]], the power series ring is a Noetherian ring. • If R is a Noetherian ring and I is a two-sided ideal, then the factor ring R/I is also Noetherian. Stated differently, the image of any surjective ring homomorphism of a Noetherian ring is Noetherian. • Every finitely-generated commutative algebra over a commutative Noetherian ring is Noetherian. (This follows from the two previous properties.) • A ring R is left-Noetherian if and only if every finitely generated left R-module is a Noetherian module. • Every localization of a commutative Noetherian ring is Noetherian. • A consequence of the Akizuki-Hopkins-Levitzki Theorem is that every left Artinian ring is left Noetherian. Another consequence is that a left Artinian ring is right Noetherian if and only if right Artinian. The analogous statements with “right” and “left” interchanged are also true. • A left Noetherian ring is left coherent and a left Noetherian domain is a left Ore domain. • A ring is (left/right) Noetherian if and only if every direct sum of injective (left/right) modules is injective. Every injective module can be decomposed as direct sum of indecomposable injective modules. • In a commutative Noetherian ring, there are only finitely many minimal prime ideals. • In a commutative Noetherian domain R, every element can be factorized into irreducible elements. Thus, if, in addition, irreducible elements are prime elements, then R is a unique factorization domain.

174.3 Examples • Any field, including fields of rational numbers, real numbers, and complex numbers, is Noetherian. (A field only has two ideals — itself and (0).) • Any principal ideal domain, such as the integers, is Noetherian since every ideal is generated by a single element. • A Dedekind domain (e.g., rings of integers) is Noetherian since every ideal is generated by at most two elements. The “Noetherian” follows from the Krull–Akizuki theorem. The bounds on the number of the generators is a corollary of the Forster–Swan theorem (or basic ring theory). • The coordinate ring of an affine variety is a Noetherian ring, as a consequence of the Hilbert basis theorem. • The enveloping algebra U of a finite-dimensional Lie algebra g is a both left and right noetherian ring; this follows from the fact that the associated graded ring of U is a quotient of Sym(g) , which is a polynomial ring over a field; thus, noetherian.[2] • The ring of polynomials in finitely-many variables over the integers or a field. Rings that are not Noetherian tend to be (in some sense) very large. Here are three examples of non-Noetherian rings: • The ring of polynomials in infinitely-many variables, X1 , X2 , X3 , etc. The sequence of ideals (X1 ), (X1 , X2 ), (X1 , X2 , X3 ), etc. is ascending, and does not terminate. • The ring of algebraic integers is not Noetherian. For example, it contains the infinite ascending chain of principal ideals: (2), (21/2 ), (21/4 ), (21/8 ), ...

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• The ring of continuous functions from the real numbers to the real numbers is not Noetherian: Let In be the ideal of all continuous functions f such that f(x) = 0 for all x ≥ n. The sequence of ideals I 0 , I 1 , I 2 , etc., is an ascending chain that does not terminate. However, a non-Noetherian ring can be a subring of a Noetherian ring. Since any integral domain is a subring of a field, any integral domain that is not Noetherian provides an example. To give a less trivial example, • The ring of rational functions generated by x and y/xn over a field k is a subring of the field k(x,y) in only two variables. Indeed, there are rings that are right Noetherian, but not left Noetherian, so that one must be careful in measuring the “size” of a ring this way. For example, if L is a subgroup of Q2 isomorphic to Z, let R be the ring of homomorphisms f from Q2 to itself satisfying f(L) ⊂ L. Choosing a basis, we can describe the same ring R as {[ R=

a 0

] } β a ∈ Z, β ∈ Q, γ ∈ Q . γ

This ring is right Noetherian, but not left Noetherian; the subset I⊂R consisting of elements with a=0 and γ=0 is a left ideal that is not finitely generated as a left R-module. If R is a commutative subring of a left Noetherian ring S, and S is finitely generated as a left R-module, then R is Noetherian.[3] (In the special case when S is commutative, this is known as Eakin’s theorem.) However this is not true if R is not commutative: the ring R of the previous paragraph is a subring of the left Noetherian ring S = Hom(Q2 ,Q2 ), and S is finitely generated as a left R-module, but R is not left Noetherian. A unique factorization domain is not necessarily a noetherian ring. It does satisfy a weaker condition: the ascending chain condition on principal ideals. A valuation ring is not Noetherian unless it is a principal ideal domain. It gives an example of a ring that arises naturally in algebraic geometry but is not Noetherian.

174.4 Primary decomposition Main article: Lasker–Noether theorem In the ring Z of integers, an arbitrary ideal is of the form (n) for some integer n (where (n) denotes the set of all integer multiples of n). If n is non-zero, and is ∏ neither 1 nor −1, by the fundamental theorem of arithmetic, there exist primes pi, and positive integers ei, with n = i pi ei . In this case, the ideal (n) may be written as the intersection of the ideals (piei ); that is, (n) = ∩i (pi ei ) . This is referred to as a primary decomposition of the ideal (n). In general, an ideal Q of a ring is said to be primary if Q is proper and whenever xy ∈ Q, either x ∈ Q or yn ∈ Q for some positive integer n. In Z, the primary ideals are precisely the ideals of the form (pe ) where p is prime and e is a positive integer. Thus, a primary decomposition of (n) corresponds to representing (n) as the intersection of finitely many primary ideals. Since the fundamental theorem of arithmetic applied to a non-zero integer n that is neither 1 nor −1 also asserts ∏ uniqueness of the representation n = i pi ei for pi prime and ei positive, a primary decomposition of (n) is essentially unique. For all of the above reasons, the following theorem, referred to as the Lasker–Noether theorem, may be seen as a certain generalization of the fundamental theorem of arithmetic: Lasker-Noether Theorem. Let R be a commutative Noetherian ring and let I be an ideal of R. Then I may be written as the intersection of finitely many primary ideals with distinct radicals; that is: I=

t ∩

Qi

i=1

with Qi primary for all i and Rad(Qi) ≠ Rad(Qj) for i ≠ j. Furthermore, if:

174.5. SEE ALSO

I=

k ∩

525

Pi

i=1

is decomposition of I with Rad(Pi) ≠ Rad(Pj) for i ≠ j, and both decompositions of I are irredundant (meaning that no proper subset of either {Q1 , ..., Qt} or {P 1 , ..., Pk} yields an intersection equal to I), t = k and (after possibly renumbering the Qi) Rad(Qi) = Rad(Pi) for all i. For any primary decomposition of I, the set of all radicals, that is, the set {Rad(Q1 ), ..., Rad(Qt)} remains the same by the Lasker–Noether theorem. In fact, it turns out that (for a Noetherian ring) the set is precisely the assassinator of the module R/I; that is, the set of all annihilators of R/I (viewed as a module over R) that are prime.

174.5 See also • Krull–Akizuki theorem • Noetherian scheme • Artinian ring • Artin–Rees lemma • Krull’s principal ideal theorem

174.6 References [1] Lam (2001), p. 19 [2] Bourbaki 1989, Ch III, §2, no. 10, Remarks at the end of the number [3] Formanek & Jategaonkar 1974, Theorem 3

• Nicolas Bourbaki, Commutative algebra • Formanek, Edward; Jategaonkar, Arun Vinayak (1974). “Subrings of Noetherian rings”. Proc. Amer. Math. Soc. 46 (2): 181–186. doi:10.2307/2039890. • Lam, T.Y. (2001). A first course in noncommutative rings. New York: Springer. p. 19. ISBN 0387951830. • Chapter X of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-201-55540-0, Zbl 0848.13001

174.7 External links • Hazewinkel, Michiel, ed. (2001), “Noetherian ring”, Encyclopedia of Mathematics, Springer, ISBN 978-155608-010-4

Chapter 175

Noncommutative ring In mathematics, more specifically modern algebra and ring theory, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exists a and b in R with a·b ≠ b·a. Noncommutative algebra is the study of results applying to rings that are not required to be commutative; however, many important results in this area apply to commutative rings as special cases.[1]

175.1 Examples Some examples of rings which are not commutative follow: • the matrix ring of n-by-n matrices over the real numbers, where n>1. • Hamilton’s quaternions. • any group algebra made from a group that is not abelian.

175.2 History Beginning with division rings arising from geometry, the study of noncommutative rings has grown into a major area of modern algebra. The theory and exposition of noncommutative rings was expanded and refined in the 19th and 20th centuries by numerous authors. An incomplete list of such contributors includes E. Artin, Richard Brauer, P. M. Cohn, W. R. Hamilton, I. N. Herstein, N. Jacobson, K. Morita, E. Noether, Ø. Ore and others.

175.3 Differences between commutative and noncommutative algebra Because noncommutative rings are a much larger class of rings than the commutative rings, their structure and behavior is less well understood. A great deal of work has been done successfully generalizing some results from commutative rings to noncommutative rings. A major difference between rings which are and are not commutative is the necessity to consider right ideals and left ideals. It is common for noncommutative ring theorists to enforce a condition on one of these types of ideals while not requiring it to hold for the opposite side. For commutative rings, the left-right distinction does not exist.

175.4 Important classes of noncommutative rings 175.4.1

Division rings

Main article: Division ring

526

175.5. IMPORTANT THEOREMS

527

A division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nonzero ring[2] in which every nonzero element a has a multiplicative inverse, i.e., an element x with a·x = x·a = 1. Stated differently, a ring is a division ring if and only if the group of units equals the set of all nonzero elements. Division rings differ from fields only in that their multiplication is not required to be commutative. However, by Wedderburn’s little theorem all finite division rings are commutative and therefore finite fields. Historically, division rings were sometimes referred to as fields, while fields were called “commutative fields”.

175.4.2

Semisimple rings

Main article: Semisimple ring A module over a (not necessarily commutative) ring with unity is said to be semisimple (or completely reducible) if it is the direct sum of simple (irreducible) submodules. A ring is said to be (left)-semisimple if it is semisimple as a left module over itself. Surprisingly, a left-semisimple ring is also right-semisimple and vice versa. The left/right distinction is therefore unnecessary.

175.4.3

Semiprimitive rings

Main article: Semiprimitive ring In algebra, a semiprimitive ring or Jacobson semisimple ring or J-semisimple ring is a ring whose Jacobson radical is zero. This is a type of ring more general than a semisimple ring, but where simple modules still provide enough information about the ring. Rings such as the ring of integers are semiprimitive, and an artinian semiprimitive ring is just a semisimple ring. Semiprimitive rings can be understood as subdirect products of primitive rings, which are described by the Jacobson density theorem.

175.4.4

Simple rings

Main article: Simple ring A simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra. Rings which are simple as rings but not as modules do exist: the full matrix ring over a field does not have any nontrivial ideals (since any ideal of M(n,R) is of the form M(n,I) with I an ideal of R), but has nontrivial left ideals (namely, the sets of matrices which have some fixed zero columns). According to the Artin–Wedderburn theorem, every simple ring that is left or right Artinian is a matrix ring over a division ring. In particular, the only simple rings that are a finite-dimensional vector space over the real numbers are rings of matrices over either the real numbers, the complex numbers, or the quaternions. Any quotient of a ring by a maximal ideal is a simple ring. In particular, a field is a simple ring. A ring R is simple if and only its opposite ring Ro is simple. An example of a simple ring that is not a matrix ring over a division ring is the Weyl algebra.

175.5 Important theorems 175.5.1

Wedderburn’s Little Theorem

Main article: Wedderburn’s little theorem Wedderburn’s little theorem states that every finite domain is a field. In other words, for finite rings, there is no distinction between domains, skew-fields and fields. The Artin–Zorn theorem generalizes the theorem to alternative rings: every finite simple alternative ring is a field.[3]

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175.5.2

Artin-Wedderburn Theorem

Main article: Artin–Wedderburn theorem The Artin–Wedderburn theorem is a classification theorem for semisimple rings and semisimple algebras. The theorem states that an (Artinian) [4] semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely determined up to permutation of the index i. In particular, any simple left or right Artinian ring is isomorphic to an n-by-n matrix ring over a division ring D, where both n and D are uniquely determined.[5] As a direct corollary, the Artin–Wedderburn theorem implies that every simple ring that is finite-dimensional over a division ring (a simple algebra) is a matrix ring. This is Joseph Wedderburn's original result. Emil Artin later generalized it to the case of Artinian rings.

175.5.3

Jacobson density theorem

Main article: Jacobson density theorem the Jacobson density theorem is a theorem concerning simple modules over a ring R.[6] The theorem can be applied to show that any primitive ring can be viewed as a “dense” subring of the ring of linear transformations of a vector space.[7][8] This theorem first appeared in the literature in 1945, in the famous paper “Structure Theory of Simple Rings Without Finiteness Assumptions” by Nathan Jacobson.[9] This can be viewed as a kind of generalization of the Artin-Wedderburn theorem's conclusion about the structure of simple Artinian rings. More formally, the theorem can be stated as follows: The Jacobson Density Theorem. Let U be a simple right R-module, D = End(UR), and X ⊂ U a finite and D-linearly independent set. If A is a D-linear transformation on U then there exists r ∈ R such that A(x) = x • r for all x in X.[10]

175.5.4

Nakayama’s lemma

Main article: Nakayama’s lemma A version of the lemma holds for right modules over non-commutative unitary rings R. The resulting theorem is sometimes known as the Jacobson–Azumaya theorem.[11] Let J(R) be the Jacobson radical of R. If U is a right module over a ring, R, and I is an right ideal in R, then define U·I to be the set of all (finite) sums of elements of the form u·i, where · is simply the action of R on U. Necessarily, U·I is a submodule of U. If V is a maximal submodule of U, then U/V is simple. So U·J(R) is necessarily a subset of V, by the definition of J(R) and the fact that U/V is simple.[12] Thus, if U contains at least one (proper) maximal submodule, U·J(R) is a proper submodule of U. However, this need not hold for arbitrary modules U over R, for U need not contain any maximal submodules.[13] Naturally, if U is a Noetherian module, this holds. If R is Noetherian, and U is finitely generated, then U is a Noetherian module over R, and the conclusion is satisfied.[14] Somewhat remarkable is that the weaker assumption, namely that U is finitely generated as an R-module (and no finiteness assumption on R), is sufficient to guarantee the conclusion. This is essentially the statement of Nakayama’s lemma.[15] Precisely, one has: Nakayama’s lemma: Let U be a finitely generated right module over a ring R. If U is a non-zero module, then U·J(R) is a proper submodule of U.[15]

175.5.5

Noncommutative localization

Main article: Localization of a ring

175.5. IMPORTANT THEOREMS

529

Localization is a systematic method of adding multiplicative inverses to a ring, and is usually applied to commutative rings. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units (invertible elements) in R*. Further one wants R* to be the 'best possible' or 'most general' way to do this – in the usual fashion this should be expressed by a universal property. The localization of R by S is usually denoted by S −1 R; however other notations are used in some important special cases. If S is the set of the non zero elements of an integral domain, then the localization is the field of fractions and thus usually denoted Frac(R). Localizing non-commutative rings is more difficult; the localization does not exist for every set S of prospective units. One condition which ensures that the localization exists is the Ore condition. One case for non-commutative rings where localization has a clear interest is for rings of differential operators. It has the interpretation, for example, of adjoining a formal inverse D−1 for a differentiation operator D. This is done in many contexts in methods for differential equations. There is now a large mathematical theory about it, named microlocalization, connecting with numerous other branches. The micro- tag is to do with connections with Fourier theory, in particular.

175.5.6

Morita equivalence

Main article: Morita equivalence Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. It is named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958. Two rings R and S (associative, with 1) are said to be (Morita) equivalent if there is an equivalence of the category of (left) modules over R, R-Mod, and the category of (left) modules over S, S-Mod. It can be shown that the left module categories R-Mod and S-Mod are equivalent if and only if the right module categories Mod-R and Mod-S are equivalent. Further it can be shown that any functor from R-Mod to S-Mod that yields an equivalence is automatically additive.

175.5.7

Brauer group

Main article: Brauer group The Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is named after the algebraist Richard Brauer. The group may also be defined in terms of Galois cohomology. More generally, the Brauer group of a scheme is defined in terms of Azumaya algebras.

175.5.8

Ore conditions

Main article: Ore condition The Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more generally localization of a ring. The right Ore condition for a multiplicative subset S of a ring R is that for a ∈ R and s ∈ S, the intersection aS ∩ sR ≠ ∅.[16] A domain that satisfies the right Ore condition is called a right Ore domain. The left case is defined similarly.

175.5.9

Goldie’s theorem

Main article: Goldie’s theorem In mathematics, Goldie’s theorem is a basic structural result in ring theory, proved by Alfred Goldie during the 1950s. What is now termed a right Goldie ring is a ring R that has finite uniform dimension (="finite rank”) as a

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CHAPTER 175. NONCOMMUTATIVE RING

right module over itself, and satisfies the ascending chain condition on right annihilators of subsets of R. Goldie’s theorem states that the semiprime right Goldie rings are precisely those that have a semisimple Artinian right classical ring of quotients. The structure of this ring of quotients is then completely determined by the Artin– Wedderburn theorem. In particular, Goldie’s theorem applies to semiprime right Noetherian rings, since by definition right Noetherian rings have the ascending chain condition on all right ideals. This is sufficient to guarantee that a right-Noetherian ring is right Goldie. The converse does not hold: every right Ore domain is a right Goldie domain, and hence so is every commutative integral domain. A consequence of Goldie’s theorem, again due to Goldie, is that every semiprime principal right ideal ring is isomorphic to a finite direct sum of prime principal right ideal rings. Every prime principal right ideal ring is isomorphic to a matrix ring over a right Ore domain.

175.6 See also • Noncommutative harmonic analysis • Representation theory (group theory) • Derived algebraic geometry • Noncommutative algebraic geometry

175.7 References [1] Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249, ISBN 978-0-387-97527-6 [2] In this article, rings have a 1. [3] Shult, Ernest E. (2011). Points and lines. Characterizing the classical geometries. Universitext. Berlin: Springer-Verlag. p. 123. ISBN 978-3-642-15626-7. Zbl 1213.51001. [4] Semisimple rings are necessarily Artinian rings. Some authors use “semisimple” to mean the ring has a trivial Jacobson radical. For Artinian rings, the two notions are equivalent, so “Artinian” is included here to eliminate that ambiguity. [5] John A. Beachy (1999). Introductory Lectures on Rings and Modules. Cambridge University Press. p. 156. ISBN 978-0521-64407-5. [6] Isaacs, p. 184 [7] Such rings of linear transformations are also known as full linear rings. [8] Isaacs, Corollary 13.16, p. 187 [9] Jacobson, Nathan “Structure Theory of Simple Rings Without Finiteness Assumptions” [10] Isaacs, Theorem 13.14, p. 185 [11] Nagata 1962, §A2 [12] Isaacs 1993, p. 182 [13] Isaacs 1993, p. 183 [14] Isaacs 1993, Theorem 12.19, p. 172 [15] Isaacs 1993, Theorem 13.11, p. 183 [16] Cohn, P. M. (1991). “Chap. 9.1”. Algebra. Vol. 3 (2nd ed.). p. 351.

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175.8 Further reading • Isaacs, I. Martin (1993). Algebra, a graduate course (1st ed.). Brooks/Cole Publishing Company. ISBN 0534-19002-2. • Herstein, I. N. (1968). Noncommutative rings (1st ed.). The Mathematical Association of America. ISBN 0-88385-015-X.

Chapter 176

Noncommutative unique factorization domain In mathematics, the noncommutative unique factorization domain is the noncommutative counterpart of the commutative or classical unique factorization domain (UFD).

176.1 Example • The ring of integral quaternions. If the coefficients a0 , a1 , a2 , a3 are integers or halves of odd integers of a rational quaternion a = a0 + a1 i + a2 j + a3 k then the quaternion is integral.

176.2 References • “Certain number-theoretic episodes in algebra”, R. Sivaramakrishnan; Publisher CRC Press, 2006, ISBN 08247-5895-1

176.3 Notes

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Chapter 177

Novikov ring For a concept in quantum cohomology, see the linked article. In mathematics, given an ∑ additive subgroup Γ ⊂ R , the Novikov ring Nov(Γ) of Γ is the subring of Z[[Γ]] [1] consisting of formal sums nγi tγi such that γ1 > γ2 > · · · and γi → −∞ . The notion was introduced by S. P. Novikov in the papers that initiated the generalization of Morse theory using a closed one-form instead of a function. The Novikov ring Nov(Γ) is a principal ideal domain. Let S be the subset of Z[Γ] consisting of those with leading term 1. Since the elements of S are unit elements of Nov(Γ) , the localization Nov(Γ)[S −1 ] of Nov(Γ) with respect to S is a subring of Nov(Γ) called the “rational part” of Nov(Γ) ; it is also a principal ideal domain.

177.1 Novikov numbers Given a smooth function f on a smooth manifold M with nondegenerate critical points, the usual Morse theory constructs a free chain complex C∗ (f ) such that the (integral) rank of Cp is the number of critical points of f of index p (called the Morse number). It computes the homology of M: H ∗ (C∗ (f )) ≈ H ∗ (M, Z) (cf. Morse homology.) In an analogy with this, one can define “Novikov numbers”. Let X be a connected polyhedron with a base point. Each cohomology class ξ ∈ H 1 (X, R) may be viewed as a linear functional on the first homology group H1 (X, R) and, composed with the Hurewicz homomorphism, it can be viewed as a group homomorphism ξ : π = π1 (X) → R . By the universal property, this map in turns gives a ring homomorphism ϕξ : Z[π] → Nov = Nov(R) , making Nov a module over Z[π] . Since X is a connected polyhedron, a local coefficient system over it corresponds one-to-one to a Z[π] -module. Let Lξ be a local coefficient system corresponding to Nov with module structure given by ϕξ . The homology group Hp (X, Lξ ) is a finitely genereated module over Nov , which is, by the structure theorem, is a direct sum of the free part and the torsion part. The rank of the free part is called the Novikov Betti number and is denoted by bp (ξ) . The number of cyclic modules in the torsion part is denoted by qp (ξ) . If ξ = 0 , Lξ is trivial and bp (0) is the usual Betti number of X. The analog of Morse inequalities holds for Novikov numbers as well (cf. the reference for now.)

177.2 Notes ∑ [1] Here, Z[[Γ]] is the ring consisting of the formal sums γ∈Γ nγ tγ , nγ integers and t a formal variable, such that the multiplication is an extension of a multiplication in the integral group ring Z[Γ] .

177.3 References • Farber, Michael (2004). Topology of closed one-forms. Mathematical surveys and monographs 108. American Mathematical Society. ISBN 0-8218-3531-9. Zbl 1052.58016. • S. P. Novikov, Multi-valued functions and functionals: An analogue of Morse theory. Soviet Math. Doklady 24 (1981), 222–226. 533

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• S. P. Novikov: The Hamiltonian formalism and a multi-valued analogue of Morse theory. Russian Mathematical Surveys 35:5 (1982), 1–56.

177.4 External links • http://mathoverflow.net/questions/13203/different-definitions-of-novikov-ring

Chapter 178

Opposite ring In algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order.[1] More precisely, the opposite of a ring (R, +, ·) is the ring (R, +, *), whose multiplication '*' is defined by a * b = b · a. (Ring addition is per definition always commutative.)

178.1 Properties Two rings R1 and R2 are isomorphic if and only if their corresponding opposite rings are isomorphic. The opposite of the opposite of a ring is isomorphic to that ring. A ring and its opposite ring are anti-isomorphic. A commutative ring is always equal to its opposite ring. A non-commutative ring may or may not be isomorphic to its opposite ring.

178.2 Notes [1] Berrick & Keating (2000), p. 19

178.3 References • Berrick, A. J.; Keating, M. E. (2000). An Introduction to Rings and Modules With K-theory in View. Cambridge studies in advanced mathematics 65. Cambridge University Press. ISBN 978-0-521-63274-4.

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Chapter 179

Order (ring theory) “Maximal order” redirects here. For the maximal order of an arithmetic function, see Extremal orders of an arithmetic function. In mathematics, an order in the sense of ring theory is a subring O of a ring A , such that 1. A is a ring which is a finite-dimensional algebra over the rational number field Q 2. O spans A over Q , so that QO = A , and 3. O is a Z-lattice in A. The last two conditions condition can be stated in less formal terms: Additively, O is a free abelian group generated by a basis for A over Q . More generally for R an integral domain contained in a field K we define O to be an R-order in a K-algebra A if it is a subring of A which is a full R-lattice.[1] When A is not a commutative ring, the idea of order is still important, but the phenomena are different. For example, the Hurwitz quaternions form a maximal order in the quaternions with rational co-ordinates; they are not the quaternions with integer coordinates in the most obvious sense. Maximal orders exist in general, but need not be unique: there is in general no largest order, but a number of maximal orders. An important class of examples is that of integral group rings. Examples:[2] • If A is the matrix ring Mn(K) over K then the matrix ring Mn(R) over R is an R-order in A • If R is an integral domain and L a finite separable extension of K, then the integral closure S of R in L is an R-order in L. • If a in A is an integral element over R then the polynomial ring R[a] is an R-order in the algebra K[a] • If A is the group ring K[G] of a finite group G then R[G] is an R-order on K[G] A fundamental property of R-orders is that every element of an R-order is integral over R.[3] If the integral closure S of R in A is an R-order then this result shows that S must be the maximal R-order in A. However this is not always the case: indeed S need not even be a ring, and even if S is a ring (for example, when A is commutative) then S need not be an R-lattice.[3]

179.1 Algebraic number theory The leading example is the case where A is a number field K and O is its ring of integers. In algebraic number theory there are examples for any K other than the rational field of proper subrings of the ring of integers that are 536

179.2. SEE ALSO

537

also orders. For example in the field extension A=Q(i) of Gaussian rationals over Q, the integral closure of Z is the ring of Gaussian integers Z[i] and so this is the unique maximal Z-order: all other orders in A are contained in it: for example, we can take the subring of the

a + bi, for which b is an even number.[4] The maximal order question can be examined at a local field level. This technique is applied in algebraic number theory and modular representation theory.

179.2 See also • Hurwitz quaternion order - An example of ring order

179.3 References [1] Reiner (2003) p.108 [2] Reiner (2003) pp.108–109 [3] Reiner (2003) p.110 [4] Pohst&Zassenhaus (1989) p.22

• Pohst, M.; Zassenhaus, H. (1989). Algorithmic Algebraic Number Theory. Encyclopedia of Mathematics and its Applications 30. Cambridge University Press. ISBN 0-521-33060-2. Zbl 0685.12001. • Reiner, I. (2003). Maximal Orders. London Mathematical Society Monographs. New Series 28. Oxford University Press. ISBN 0-19-852673-3. Zbl 1024.16008.

Chapter 180

Ore algebra In computer algebra, an Ore algebra is a special kind of iterated Ore extension that can be used to represent linear functional operators, including linear differential and/or recurrence operators.[1] The concept is named after Øystein Ore.

180.1 Definition Let K be a (commutative) field and A = K[x1 , . . . , xs ] be a commutative polynomial ring (with A = K when s = 0 ). The iterated skew polynomial ring A[∂1 ; σ1 , δ1 ] · · · [∂r ; σr , δr ] is called an Ore algebra when the σi and δj commute for i ̸= j , and satisfy σi (∂j ) = ∂j , δi (∂j ) = 0 for i > j .

180.2 Properties Ore algebras satisfy the Ore condition, and thus can be embedded in a (skew) field of fractions. The constraint of commutation in the definition makes Ore algebras have a non-commutative generalization theory of Gröbner basis for their left ideals.

180.3 References [1] Chyzak, Frédéric; Salvy, Bruno (1998). “Non-commutative Elimination in Ore Algebras Proves Multivariate Identities”. Journal of Symbolic Computation (Elsevier) 26 (2): 187–227. doi:10.1006/jsco.1998.0207.

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Chapter 181

Ore condition For Ore’s condition in graph theory, see Ore’s theorem. In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more generally localization of a ring. The right Ore condition for a multiplicative subset S of a ring R is that for a ∈ R and s ∈ S, the intersection aS ∩ sR ≠ ∅.[1] A domain that satisfies the right Ore condition is called a right Ore domain. The left case is defined similarly.

181.1 General idea The goal is to construct the right ring of fractions R[S −1 ] with respect to multiplicative subset S. In other words we want to work with elements of the form as−1 and have a ring structure on the set R[S −1 ]. The problem is that there is no obvious interpretation of the product (as−1 )(bt −1 ); indeed, we need a method to “move” s−1 past b. This means that we need to be able to rewrite s−1 b as a product b1 s1 −1 .[2] Suppose s−1 b = b1 s1 −1 then multiplying on the left by s and on the right by s1 , we get bs1 = sb1 . Hence we see the necessity, for a given a and s, of the existence of a1 and s1 with s1 ≠ 0 and such that as1 = sa1 .

181.2 Application Since it is well known that each integral domain is a subring of a field of fractions (via an embedding) in such a way that every element is of the form rs−1 with s nonzero, it is natural to ask if the same construction can take a noncommutative domain and associate a division ring (a noncommutative field) with the same property. It turns out that the answer is sometimes “no”, that is, there are domains which do not have an analogous “right division ring of fractions”. For every right Ore domain R, there is a unique (up to natural R-isomorphism) division ring D containing R as a subring such that every element of D is of the form rs−1 for r in R and s nonzero in R. Such a division ring D is called a ring of right fractions of R, and R is called a right order in D. The notion of a ring of left fractions and left order are defined analogously, with elements of D being of the form s−1 r. It is important to remember that the definition of R being a right order in D includes the condition that D must consist entirely of elements of the form rs−1 . Any domain satisfying one of the Ore conditions can be considered a subring of a division ring, however this does not automatically mean R is a left order in D, since it is possible D has an element which is not of the form s−1 r. Thus it is possible for R to be a right-not-left Ore domain. Intuitively, the condition that all elements of D be of the form rs−1 says that R is a “big” R-submodule of D. In fact the condition ensures RR is an essential submodule of DR. Lastly, there is even an example of a domain in a division ring which satisfies neither Ore condition (see examples below). Another natural question is: “When is a subring of a division ring right Ore?" One characterization is that a subring R of a division ring D is a right Ore domain if and only if D is a flat left R-module (Lam 2007, Ex. 10.20). 539

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A different, stronger version of the Ore conditions is usually given for the case where R is not a domain, namely that there should be a common multiple c = au = bv with u, v not zero divisors. In this case, Ore’s theorem guarantees the existence of an over-ring called the (right or left) classical ring of quotients.

181.3 Examples Commutative domains are automatically Ore domains, since for nonzero a and b, ab is nonzero in aR ∩ bR. Right Noetherian domains, such as right principal ideal domains, are also known to be right Ore domains. Even more generally, Alfred Goldie proved that a domain R is right Ore if and only if RR has finite uniform dimension. It is also true that right Bézout domains are right Ore. A subdomain of a division ring which is not right or left Ore: If F is any field, and G = ⟨x, y⟩ is the free monoid on two symbols x and y, then the monoid ring F [G] does not satisfy any Ore condition, but it is a free ideal ring and thus indeed a subring of a division ring, by (Cohn 1995, Cor 4.5.9).

181.4 Multiplicative sets The Ore condition can be generalized to other multiplicative subsets, and is presented in textbook form in (Lam 1999, §10) and (Lam 2007, §10). A subset S of a ring R is called a right denominator set if it satisfies the following three conditions for every a,b in R, and s, t in S: 1. st in S; (The set S is multiplicatively closed.) 2. aS ∩ sR is not empty; (The set S is right permutable.) 3. If sa = 0, then there is some u in S with au = 0; (The set S is right reversible.) If S is a right denominator set, then one can construct the ring of right fractions RS −1 similarly to the commutative case. If S is taken to be the set of regular elements (those elements a in R such that if b in R is nonzero, then ab and ba are nonzero), then the right Ore condition is simply the requirement that S be a right denominator set. Many properties of commutative localization hold in this more general setting. If S is a right denominator set for a ring R, then the left R-module RS −1 is flat. Furthermore, if M is a right R-module, then the S-torsion, torS(M) = { m in M : ms = 0 for some s in S }, is an R-submodule isomorphic to Tor1 (M,RS −1 ), and the module M ⊗R RS −1 is naturally isomorphic to a module MS −1 consisting of “fractions” as in the commutative case.

181.5 Notes [1] Cohn, P. M. (1991). “Chap. 9.1”. Algebra. Vol. 3 (2nd ed.). p. 351. [2] Artin, Michael (1999). “Noncommutative Rings”. p. 13. Retrieved 9 May 2012.

181.6 References • Cohn, P. M. (1991), Algebra, Vol. 3 (2nd ed.), Chichester: John Wiley & Sons, pp. xii+474, ISBN 0-47192840-2, MR 1098018, Zbl 0719.00002 • Cohn, P.M. (1961), “On the embedding of rings in skew fields”, Proc. London Math. Soc. 11: 511–530, doi:10.1112/plms/s3-11.1.511, MR 25#100, Zbl 0104.03203

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• Cohn, P. M. (1995), Skew fields, Theory of general division rings, Encyclopedia of Mathematics and Its Applications 57, Cambridge University Press, ISBN 0-521-43217-0, Zbl 0840.16001 • Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, Zbl 0911.16001 • Lam, Tsit-Yuen (2007), Exercises in modules and rings, Problem Books in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98850-4, MR 2278849, Zbl 1121.16001 • Stenström, Bo (1971), Rings and modules of quotients, Lecture Notes in Mathematics 237, Berlin: SpringerVerlag, pp. vii+136, doi:10.1007/BFb0059904, ISBN 978-3-540-05690-4, MR 0325663, Zbl 0229.16003

181.7 External links • PlanetMath page on Ore condition • PlanetMath page on Ore’s theorem • PlanetMath page on classical ring of quotients

Chapter 182

Ore extension In mathematics, especially in the area of algebra known as ring theory, an Ore extension, named after Øystein Ore, is a special type of a ring extension whose properties are relatively well understood. Ore extensions appear in several natural contexts, including skew and differential polynomial rings, group algebras of polycyclic groups, universal enveloping algebras of solvable Lie algebras, and coordinate rings of quantum groups.

182.1 Definition Suppose that R is a (not necessarily commutative) ring, σ:R → R is a ring homomorphism, and δ:R → R is a σderivation of R, which means that δ is a homomorphism of abelian groups satisfying

δ(r1 r2 ) = σ(r1 )δ(r2 ) + δ(r1 )r2 . Then the Ore extension R[x;σ,δ], also called a skew polynomial ring, is the noncommutative ring obtained by giving the ring of polynomials R[x] a new multiplication, subject to the identity

xr = σ(r)x + δ(r). If δ = 0 (i.e., is the zero map) then the Ore extension is denoted R[x; σ]. If σ = 1 (i.e., the identity map) then the Ore extension is denoted R[x,δ] and is called a differential polynomial ring.

182.2 Examples The Weyl algebras are Ore extensions, with R any a commutative polynomial ring, σ the identity ring endomorphism, and δ the polynomial derivative. Ore algebras are a class of iterated Ore extensions under suitable constraints that permit to develop a noncommutative extension of the theory of Gröbner bases.

182.3 Properties • An Ore extension of a domain is a domain. • An Ore extension of a skew field is a non-commutative Principal ideal domain. • If σ is an automorphism and R is a left Noetherian ring then the Ore extension R[λ;σ,δ] is also left Noetherian. 542

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182.4 Elements An element f of an Ore ring R is called • twosided[1] (or invariant[2] ), if R·f = f·R, and • central, if g·f = f·g for all g ∈ R.

182.5 Further reading • Goodearl, K. R.; Warfield, R. B., Jr. (2004), An Introduction to Noncommutative Noetherian Rings, Second Edition, London Mathematical Society Student Texts 61, Cambridge: Cambridge University Press, ISBN 0521-54537-4, MR 2080008 • McConnell, J. C.; Robson, J. C. (2001), Noncommutative Noetherian rings, Graduate Studies in Mathematics 30, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2169-5, MR 1811901 • Rowen, Louis H. (1988), Ring theory, vol. I, II, Pure and Applied Mathematics, 127, 128, Boston, MA: Academic Press, ISBN 0-12-599841-4, MR 940245

182.6 References [1] Jacobson, Nathan (1996). Finite-Dimensional Division Algebras over Fields. Springer. [2] Cohn, P. M. (1995). Skew Fields: Theory of General Division Rings. Cambridge University Press.

Chapter 183

Outermorphism This article is about outermorphisms of Clifford algebras over a finite-dimensional inner product space. For the concept in group theory, see outer automorphism group. In geometric algebra, the outermorphism of a linear function between vectors is a natural extension of the map to arbitrary multivectors.[1]

183.1 Definition Let f be an R-linear map from V to W. The outermorphism of f is the unique map f : Λ(V) → Λ(W) satisfying

f(x) = f (x) f(A ∧ B) = f(A) ∧ f(B) f(A + B) = f(A) + f(B) f(1) = 1 for all vectors x and all multivectors A and B, where Λ(V) denotes the exterior algebra over V. The outermorphism inherits linearity properties of the original linear map. For example, we see that for scalars α, β and vectors x, y, z, the outermorphism is linear over bivectors:

f(αx ∧ z + βy ∧ z) = f((αx + βy) ∧ z) = f (αx + βy) ∧ f (z) = (αf (x) + βf (y)) ∧ f (z) = α(f (x) ∧ f (z)) + β(f (y) ∧ f (z)) = α f(x ∧ z) + β f(y ∧ z), which extends through the axiom of distributivity over addition above to linearity over all multivectors.

183.2 Adjoint Let a and b be vectors and f be an outermorphism. We define the adjoint function f to be the function that satisfies the property 544

183.3. PROPERTIES

545

b · f(a) = a · f(b). If geometric calculus is available, then the adjoint may be extracted more directly:

f(a) = ∇b ⟨af(b)⟩ Note that the above definition of adjoint is like the definition of the transpose in matrix theory. The adjoint is itself an outermorphism. When the context is clear, the underline below the function is often omitted.

183.3 Properties It follows from the definition at the beginning that the outermorphism of a multivector A is grade-preserving:[2] ⟨ ⟩ f (⟨A⟩r ) = f (A) r where the notation ⟨ ⟩r indicates the r-vector part of A. Since any vector x may be written as x = 1 ∧ x , it follows that scalars are unaffected with f(1) = 1 . Similarly, since there is only ever one independent pseudoscalar, we must have f(I) ∝ I . The determinant is defined to be the proportionality factor:[3]

det f = f(I)I −1 The underline is not necessary in this context because the determinant of a function is the same as the determinant of its adjoint. The determinant of the composition of functions is the product of the determinants:

det(f ◦ g) = det f det g If the determinant of a function is nonzero, then the function has an inverse given by

f−1 (X) =

f(XI)I −1 = f(XI)[f(I)]−1 , det f

and so does its adjoint, with

f

−1

(X) =

I −1 f(IX) = [f(I)]−1 f(IX). det f

The concepts of eigenvalues and eigenvectors are somewhat modified. Let λ be a real number and let B be a blade of grade r. We say that B is an eigenblade of the function if

f(B) = λB, and λ is its eigenvalue. It may seem strange to consider only real eigenvalues. After all, it is widely known in linear algebra that the eigenvalues of a matrix with all real entries can have complex eigenvalues. In geometric algebra, however, the blades of different orders inherit the complex structure. Since both vectors and pseudovectors can act as eigenblades, they may each have a set of eigenvalues matching the degrees of freedom of the complex eigenvalues that would be found in ordinary linear algebra.

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183.4 Examples Simple maps The identity map and the scalar projection operator are outermorphisms. Versors A rotation of a vector by rotor R is given by f (x) = RxR† with outermorphism f(X) = RXR† . We check that this is the correct form of the outermorphism. Since rotations are built from the geometric product, which has the distributive property, they must be linear. To see that rotations are also outermorphisms, we recall that rotations preserve angles between vectors:[4] x · y = (RxR† ) · (RyR† ) Next, we try inputting a higher grade element and check that it is consistent with the original rotation for vectors: Nonexample – grade projection An example of a multivector-valued function of multivectors that is linear but is not an outermorphism is grade projection where the grade is nonzero, for example projection onto grade 1: ⟨(x ∧ y)⟩1 = 0 ⟨x⟩1 ∧ ⟨y⟩1 = x ∧ y

183.5 References [1] L. Dorst, C.J.L. Doran, J. Lasenby (2001). Applications of geometric algebra in computer science and engineering. Springer. p. 61. ISBN 0-817-642-676. [2] D. Hestenes, G. Sobczyk (1987). Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. Fundamental Theories of Physics 5. Springer. p. 68. ISBN 9-02772-5616. [3] D. Hestenes, G. Sobczyk (1987). Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. Fundamental Theories of Physics 5. Springer. p. 70. ISBN 9-02772-5616. [4] C. Perwass (2008). Geometric Algebra with Applications in Engineering. Geometry and Computing 4. Springer. p. 23. ISBN 354-089-067-X.

• W.E. Baylis (1996). Clifford (Geometric) Algebras: With Applications in Physics, Mathematics, and Engineering. Springer. p. 71. ISBN 0-817-638-687. • A. Crumeyrolle, R. Ablamowicz, P. Lounesto (1995). Clifford Algebras and Spinor Structures: A Special Volume Dedicated to the Memory of Albert Crumeyrolle (1919-−1992). Mathematics and Its Applications 321. Springer. p. 105. ISBN 0-792-333-667. • C. D'Orangeville, A. Anthony, N. Lasenby (2003). Geometric Algebra For Physicists. Cambridge University Press. p. 343. ISBN 0-521-480-221. • P. Joot. Exploring physics with Geometric Algebra. p. 157.

183.6. EXTERNAL LINKS

183.6 External links

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Chapter 184

Overring In mathematics, an overring B of an integral domain A is a subring of the field of fractions K of A that contains A: i.e., A ⊆ B ⊆ K .[1] For instance, an overring of the integers is a ring in which all elements are rational numbers, such as the ring of dyadic rationals. A typical example is given by localization: if S is a multiplicatively closed subset of A, then the localization S −1 A is an overring of A. The rings in which every overring is a localization are said to have the QR property; they include the Bézout domains and are a subset of the Prüfer domains.[2] In particular, every overring of the ring of integers arises in this way; for instance, the dyadic rationals are the localization of the integers by the powers of two.

184.1 References [1] Fontana, Marco; Papick, Ira J. (2002), “Dedekind and Prüfer domains”, in Mikhalev, Alexander V.; Pilz, Günter F., The concise handbook of algebra, Kluwer Academic Publishers, Dordrecht, pp. 165–168. [2] Fuchs, Laszlo; Heinzer, William; Olberding, Bruce (2004), “Maximal prime divisors in arithmetical rings”, Rings, modules, algebras, and abelian groups, Lecture Notes in Pure and Appl. Math. 236, Dekker, New York, pp. 189–203, MR 2050712. See in particular p. 196.

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Chapter 185

Pairing This article is about the mathematics concept. For other uses, see Pair (disambiguation). The concept of pairing treated here occurs in mathematics.

185.1 Definition Let R be a commutative ring with unity, and let M, N and L be three R-modules. A pairing is any R-bilinear map e : M × N → L . That is, it satisfies

e(rm, n) = e(m, rn) = re(m, n) e(m1 + m2 , n) = e(m1 , n) + e(m2 , n) and e(m, n1 + n2 ) = e(m, n1 ) + e(m, n2 ) for any r ∈ R and any m, m1 , m2 ∈ M and any n, n1 , n2 ∈ N . Or equivalently, a pairing is an R-linear map

M ⊗R N → L where M ⊗R N denotes the tensor product of M and N. A pairing can also be considered as an R-linear map Φ : M → HomR (N, L) , which matches the first definition by setting Φ(m)(n) := e(m, n) . A pairing is called perfect if the above map Φ is an isomorphism of R-modules. If N = M a pairing is called alternating if for the above map we have e(m, m) = 0 . A pairing is called non-degenerate if for the above map we have that e(m, n) = 0 for all m implies n = 0 .

185.2 Examples Any scalar product on a real vector space V is a pairing (set M = N = V, R = R in the above definitions). The determinant map (2 × 2 matrices over k) → k can be seen as a pairing k 2 × k 2 → k . The Hopf map S 3 → S 2 written as h : S 2 × S 2 → S 2 is an example of a pairing. In [1] for instance, Hardie et al. present an explicit construction of the map using poset models. 549

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185.3 Pairings in cryptography Main article: Pairing-based cryptography In cryptography, often the following specialized definition is used:[2] Let G1 , G2 be additive groups and GT a multiplicative group, all of prime order p . Let P ∈ G1 , Q ∈ G2 be generators of G1 and G2 respectively. A pairing is a map: e : G1 × G2 → GT for which the following holds: ) ( ab 1. Bilinearity: ∀a, b ∈ Z∗p : e P a , Qb = e (P, Q) 2. Non-degeneracy: e (P, Q) ̸= 1 3. For practical purposes, e has to be computable in an efficient manner Note that is also common in cryptographic literature for all groups to be written in multiplicative notation. In cases when G1 = G2 = G , the pairing is called symmetric. If, furthermore, G is cyclic, the map e will be commutative; that is, for any P, Q ∈ G , we have e(P, Q) = e(Q, P ) . This is because for a generator g ∈ G , there exist integers p , q such that P = g p and Q = g q . Therefore e(P, Q) = e(g p , g q ) = e(g, g)pq = e(g q , g p ) = e(Q, P ) . The Weil pairing is an important pairing in elliptic curve cryptography; e.g., it may be used to attack certain elliptic curves (see MOV attack). It and other pairings have been used to develop identity-based encryption schemes.

185.4 Slightly different usages of the notion of pairing Scalar products on complex vector spaces are sometimes called pairings, although they are not bilinear. For example, in representation theory, one has a scalar product on the characters of complex representations of a finite group which is frequently called character pairing.

185.5 References [1] A nontrivial pairing of finite T0 spaces Authors: Hardie K.A.1; Vermeulen J.J.C.; Witbooi P.J. Source: Topology and its Applications, Volume 125, Number 3, 20 November 2002 , pp. 533-542(10) [2] Dan Boneh, Matthew K. Franklin, Identity-Based Encryption from the Weil Pairing Advances in Cryptology - Proceedings of CRYPTO 2001 (2001)

185.6 External links • The Pairing-Based Crypto Lounge

Chapter 186

Paravector The name paravector is used for the sum of a scalar and a vector in any Clifford algebra (Clifford algebra is also known as geometric algebra in the physics community.) This name was given by J. G. Maks, Doctoral Dissertation, Technische Universiteit Delft (Netherlands), 1989. The complete algebra of paravectors along with corresponding higher grade generalizations, all in the context of the Euclidean space of three dimensions, is an alternative approach to the spacetime algebra (STA) introduced by David Hestenes. This alternative algebra is called algebra of physical space (APS).

186.1 Fundamental axiom For Euclidean spaces, the fundamental axiom indicates that the product of a vector with itself is the scalar value of the length squared (positive)

vv = v · v Writing

v = u + w, and introducing this into the expression of the fundamental axiom

(u + w)2 = uu + uw + wu + ww, we get the following expression after appealing to the fundamental axiom again

u · u + 2u · w + w · w = u · u + uw + wu + w · w, which allows to identify the scalar product of two vectors as

u·w=

1 (uw + wu) . 2

As an important consequence we conclude that two orthogonal vectors (with zero scalar product) anticommute

uw + wu = 0 551

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186.2 The Three-dimensional Euclidean space The following list represents an instance of a complete basis for the Cℓ3 space, {1, {e1 , e2 , e3 }, {e23 , e31 , e12 }, e123 }, which forms an eight-dimensional space, where the multiple indices indicate the product of the respective basis vectors, for example e23 = e2 e3 . The grade of a basis element is defined in terms of the vector multiplicity, such that According to the fundamental axiom, two different basis vectors anticommute,

ei ej + ej ei = 2δij or in other words,

ei ej = −ej ei ; i ̸= j This means that the volume element e123 squares to −1

e2123 = e1 e2 e3 e1 e2 e3 = e2 e3 e2 e3 = −e3 e3 = −1. Moreover, the volume element e123 commutes with any other element of the Cℓ(3) algebra, so that it can be identified with the complex number i , whenever there is no danger of confusion. In fact, the volume element e123 along with the real scalar forms an algebra isomorphic to the standard complex algebra. The volume element can be used to rewrite an equivalent form of the basis as

186.2.1

Paravectors

The corresponding paravector basis that combines a real scalar and vectors is {1, e1 , e2 , e3 } , which forms a four-dimensional linear space. The paravector space in the three-dimensional Euclidean space Cℓ3 can be used to represent the space-time of special relativity as expressed in the algebra of physical space (APS). It is convenient to write the unit scalar as 1 = e0 , so that the complete basis can be written in a compact form as {eµ }, where the Greek indices such as µ run from 0 to 3 .

186.2.2

Antiautomorphism

Reversion conjugation The Reversion antiautomorphism is denoted by † . The action of this conjugation is to reverse the order of the geometric product (product between Clifford numbers in general). (AB)† = B † A† , where vectors and real scalar numbers are invariant under reversion conjugation and are said to be real, for example: a† = a 1† = 1 On the other hand,the trivector and bivectors change sign under reversion conjugation and are said to be purely imaginary. The reversion conjugation applied to each basis element is given below

186.2. THE THREE-DIMENSIONAL EUCLIDEAN SPACE

553

Clifford conjugation The Clifford Conjugation is denoted by a bar over the object¯. This conjugation is also called bar conjugation. Clifford conjugation is the combined action of grade involution and reversion. The action of the Clifford conjugation on a paravector is to reverse the sign of the vectors, maintaining the sign of the real scalar numbers, for example ¯a = −a ¯1 = 1 This is due to both scalars and vectors being invariant to reversion ( it is impossible to reverse the order of one or no things ) and scalars are of zero order and so are of even grade whilst vectors are of odd grade and so undergo a sign change under grade involution. As antiautomorphism, the Clifford conjugation is distributed as AB = B A The bar conjugation applied to each basis element is given below • Note.- The volume element is invariant under the bar conjugation.

186.2.3

Grade automorphism †

†

†

The grade automorphism AB = A B is defined as the composite action of both the reversion conjugation and Clifford conjugation and has the effect to invert the sign of odd-grade multivectors, while maintaining the even-grade multivectors invariant:

186.2.4

Invariant subspaces according to the conjugations

Four special subspaces can be defined in the Cℓ3 space based on their symmetries under the reversion and Clifford conjugation • Scalar subspace: Invariant under Clifford conjugation. • Vector subspace: Reverses sign under Clifford conjugation. • Real subspace: Invariant under reversion conjugation. • Imaginary subspace: Reverses sign under reversion conjugation. Given p as a general Clifford number, the complementary scalar and vector parts of p are given by symmetric and antisymmetric combinations with the Clifford conjugation ⟨p⟩S = 21 (p + p), ⟨p⟩V = 12 (p − p) . In similar way, the complementary Real and Imaginary parts of p are given by symmetric and antisymmetric combinations with the Reversion conjugation ⟨p⟩R = 21 (p + p† ), ⟨p⟩I = 12 (p − p† ) . It is possible to define four intersections, listed below

⟨p⟩RS = ⟨p⟩SR ≡ ⟨⟨p⟩R ⟩S ⟨p⟩RV = ⟨p⟩V R ≡ ⟨⟨p⟩R ⟩V ⟨p⟩IV = ⟨p⟩V I ≡ ⟨⟨p⟩I ⟩V

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CHAPTER 186. PARAVECTOR

⟨p⟩IS = ⟨p⟩SI ≡ ⟨⟨p⟩I ⟩S The following table summarizes the grades of the respective subspaces, where for example, the grade 0 can be seen as the intersection of the Real and Scalar subspaces • Remark: The term “Imaginary” is used in the context of the Cℓ3 algebra and does not imply the introduction of the standard complex numbers in any form.

186.2.5

Closed Subspaces respect to the product

There are two subspaces that are closed respect to the product. They are the scalar space and the even space that are isomorphic with the well known algebras of complex numbers and quaternions. • The scalar space made of grades 0 and 3 is isomorphic with the standard algebra of complex numbers with the identification of e123 = i • The even space, made of elements of grades 0 and 2, is isomorphic with the algebra of quaternions with the identification of −e23 = i −e31 = j −e12 = k

186.2.6

Scalar Product

Given two paravectors u and v , the generalization of the scalar product is ⟨u¯ v ⟩S . The magnitude square of a paravector u is ⟨u¯ u⟩S , which is not a definite bilinear form and can be equal to zero even if the paravector is not equal to zero. It is very suggestive that the paravector space automatically obeys the metric of the Minkowski space because ηµν = ⟨eµ¯eν ⟩S and in particular: η00 = ⟨e0¯e0 ⟩ = ⟨1(1)⟩S = 1, η11 = ⟨e1¯e1 ⟩ = ⟨e1 (−e1 )⟩S = −1, η01 = ⟨e0¯e1 ⟩ = ⟨1(−e1 )⟩S = 0.

186.2.7

Biparavectors

Given two paravectors u and v , the biparavector B is defined as: B = ⟨u¯ v ⟩V . The biparavector basis can be written as {⟨eµ¯eν ⟩V }, which contains six independent elements, including real and imaginary terms. Three real elements (vectors) as

⟨e0¯ek ⟩V = −ek ,

186.2. THE THREE-DIMENSIONAL EUCLIDEAN SPACE

555

and three imaginary elements (bivectors) as

⟨ej ¯ek ⟩V = −ejk where j, k run from 1 to 3. In the Algebra of physical space, the electromagnetic field is expressed as a biparavector as

F = E + iB , where both the electric and magnetic fields are real vectors E† = E B† = B and i represents the pseudoscalar volume element. Another example of biparavector is the representation of the space-time rotation rate that can be expressed as

W = iθj ej + η j ej , with three ordinary rotation angle variables θj and three rapidities η j .

186.2.8

Triparavectors

Given three paravectors u , v and w , the triparavector T is defined as: T = ⟨u¯ v w⟩I . The triparavector basis can be written as {⟨eµ¯eν eλ ⟩I }, but there are only four independent triparavectors, so it can be reduced to {ieρ } .

186.2.9

Pseudoscalar

The pseudoscalar basis is {⟨eµ¯eν eλ¯eρ ⟩IS }, but a calculation reveals that it contains only a single term. This term is the volume element i = e1 e2 e3 . The four grades, taken in combination of pairs generate the paravector, biparavector and triparavector spaces as shown in the next table, where for example, we see that the paravector is made of grades 0 and 1

186.2.10

Paragradient

The paragradient operator is the generalization of the gradient operator in the paravector space. The paragradient in the standard paravector basis is

∂ = e0 ∂0 − e1 ∂1 − e2 ∂2 − e3 ∂3 , which allows one to write the d'Alembert operator as

¯ S = ⟨∂ ∂⟩ ¯S □ = ⟨∂∂⟩

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CHAPTER 186. PARAVECTOR

The standard gradient operator can be defined naturally as ∇ = e1 ∂1 + e2 ∂2 + e3 ∂3 , so that the paragradient can be written as ∂ = ∂0 − ∇, where e0 = 1 . The application of the paragradient operator must be done carefully, always respecting its non-commutative nature. For example, a widely used derivative is ∂ef (x)e3 = (∂f (x))ef (x)e3 e3 , where f (x) is a scalar function of the coordinates. The paragradient is an operator that always acts from the left if the function is a scalar function. However, if the function is not scalar, the paragradient can act from the right as well. For example, the following expression is expanded as (L∂) = e0 ∂0 L + (∂1 L)e1 + (∂2 L)e2 + (∂3 L)e3

186.2.11

Null Paravectors as Projectors

Null paravectors are elements that are not necessarily zero but have magnitude identical to zero. For a null paravector p , this property necessarily implies the following identity p¯ p = 0. In the context of Special Relativity they are also called lightlike paravectors. Projectors are null paravectors of the form ˆ Pk = 1 (1 + k), 2

ˆ is a unit vector. where k A projector Pk of this form has a complementary projector P¯k ˆ P¯k = 1 (1 − k), 2

such that Pk + P¯k = 1 As projectors, they are idempotent Pk = Pk Pk = Pk Pk Pk = ... and the projection of one on the other is zero because they are null paravectors Pk P¯k = 0. The associated unit vector of the projector can be extracted as ˆ = Pk − P¯k , k ˆ is an operator with eigenfunctions Pk and P¯k , with respective eigenvalues 1 and −1 . this means that k ˆ is analytic around zero From the previous result, the following identity is valid assuming that f (k) ˆ = f (1)Pk + f (−1)P¯k . f (k) This gives origin to the pacwoman property, such that the following identities are satisfied ˆ k = f (1)Pk , f (k)P ˆ P¯k = f (−1)P¯k . f (k)

186.3. HIGHER DIMENSIONS

186.2.12

557

Null Basis for the paravector space

A basis of elements, each one of them null, can be constructed for the complete Cℓ3 space. The basis of interest is the following {P¯3 , P3 e1 , P3 , e1 P3 } so that an arbitrary paravector p = p0 e0 + p1 e1 + p2 e2 + p3 e3 can be written as p = (p0 + p3 )P3 + (p0 − p3 )P¯3 + (p1 + ip2 )e1 P3 + (p1 − ip2 )P3 e1 This representation is useful for some systems that are naturally expressed in terms of the light cone variables that are the coefficients of P3 and P¯3 respectively. Every expression in the paravector space can be written in terms of the null basis. A paravector p is in general parametrized by two real scalars numbers {u, v} and a general scalar number w (including scalar and pseudoscalar numbers) p = uP¯3 + vP3 + we1 P3 + w† P3 e1 the paragradient in the null basis is ∂ = 2P3 ∂u + 2P¯3 ∂v − 2e1 P3 ∂w† − 2P3 e1 ∂w

186.3 Higher Dimensions An n-dimensional Euclidean space allows the existence of multivectors of grade n (n-vectors). The dimension of the vector ( )space is evidently equal to n and a simple combinatorial analysis shows that ( the ) dimension of the bivector space n n is . In general, the dimension of the multivector space of grade m is and the dimension of the whole 2 m n Clifford algebra Cℓ(n) is 2 . A given multivector with homogeneous grade is either invariant or changes sign under the action of the reversion conjugation † . The elements that remain invariant are defined as Hermitian and those that change sign are defined as anti-Hermitian. Grades can thus be classified as follows:

186.4 Matrix Representation The algebra of the Cℓ(3) space is isomorphic to the Pauli matrix algebra such that ( ) ( ) ( 1 0 0 0 0 ¯ from which the null basis elements become P3 = ; P3 = ; P3 e 1 = 0 0 0 1 0

) ( ) 1 0 0 ; e 1 P3 = . 0 1 0

A general Clifford number in 3D can be written as Ψ = ψ11 P3 − ψ12 P3 e1 + ψ21 e1 P3 + ψ22 P¯3 , where the coefficients ψjk are scalar elements (including pseudoscalars). The indexes were chosen such that the representation of this Clifford number in terms of the Pauli matrices is

Ψ→

( ψ11 ψ21

186.4.1

ψ12 ψ22

)

Conjugations

The reversion conjugation ( is translated into ) the Hermitian conjugation and the bar conjugation is translated into the ψ −ψ 22 12 ¯ → following matrix: Ψ , such that the scalar part is translated as −ψ21 ψ11

558

⟨Ψ⟩S →

CHAPTER 186. PARAVECTOR

( ) ψ11 + ψ22 1 0 T r[ψ] = 12×2 0 1 2 2

The rest of the subspaces are translated as ( ⟨Ψ⟩V →

0 ψ21

ψ12 0

)

⟨Ψ⟩R →

( ∗ 1 ψ11 + ψ11 ∗ 2 ψ21 + ψ12

∗ ψ12 + ψ21 ∗ ψ22 + ψ22

⟨Ψ⟩I →

( ∗ 1 ψ11 − ψ11 ∗ 2 ψ21 − ψ12

∗ ψ12 − ψ21 ∗ ψ22 − ψ22

186.4.2

)

)

Higher Dimensions

The matrix representation of a Euclidean space in higher dimensions can be constructed in terms of the Kronecker product of the Pauli matrices, resulting in complex matrices of dimension 2n . The 4D representation could be taken as The 7D representation could be taken as

186.5 Lie algebras Clifford algebras can be used to represent any classical Lie algebra. In general it is possible to identify Lie algebras of compact groups by using anti-Hermitian elements, which can be extended to non-compact groups by adding Hermitian elements. The bivectors of an n-dimensional Euclidean space are Hermitian elements and can be used to represent the spin(n) Lie algebra. The bivectors of the three-dimensional Euclidean space form the spin(3) Lie algebra, which is isomorphic to the su(2) Lie algebra. This accidental isomorphism allows to picture a geometric interpretation of the states of the two dimensional Hilbert space by using the Bloch sphere. One of those systems is the spin 1/2 particle. The spin(3) Lie algebra can be extended by adding the three unitary vectors to form a Lie algebra isomorphic to the SL(2, C) Lie algebra, which is the double cover of the Lorentz group SO(3, 1) . This isomorphism allows the possibility to develop a formalism of special relativity based on SL(2, C) , which is carried out in the form of the algebra of physical space. There is only one additional accidental isomorphism between a spin Lie algebra and a su(N ) Lie algebra. This is the isomorphism between spin(6) and su(4) . Another interesting isomorphism exists between spin(5) and sp(4) . So, the sp(4) Lie algebra can be used to generate the U Sp(4) group. Despite that this group is smaller than the SU (4) group, it is seen to be enough to span the four-dimensional Hilbert space.

186.6 See also • Algebra of physical space • Dirac equation in the algebra of physical space

186.7. REFERENCES

559

186.7 References 186.7.1

Textbooks

• Baylis, William (2002). Electrodynamics: A Modern Geometric Approach (2nd ed.). Birkhäuser. ISBN 08176-4025-8 • Baylis, William, Clifford (Geometric) Algebras With Applications in Physics, Mathematics, and Engineering, Birkhauser (1999) • [H1999] David Hestenes: New Foundations for Classical Mechanics (Second Edition). ISBN 0-7923-5514-8, Kluwer Academic Publishers (1999) • Chris Doran and Antony Lasenby, Geometric Algebra for Physicists, Cambridge, 2003

186.7.2

Articles

• William E. Baylis, Relativity in Introductory Physics, Can. J. Phys. 82 (11), 853—873 (2004). (ArXiv: physics/0406158) • C. Doran, D. Hestenes, F. Sommen and N. Van Acker, Lie groups and spin groups, J. Math. Phys. 34 (8), 1993 • R. Cabrera, W. E. Baylis, C. Rangan, Sufficient condition for the coherent control of n-qubit systems, Phys. Rev. A, 76, 033401, 2007

Chapter 187

Partially ordered ring In abstract algebra, a partially ordered ring is a ring (A, +, · ), together with a compatible partial order, i.e. a partial order ≤ on the underlying set A that is compatible with the ring operations in the sense that it satisfies: x ≤ y implies x + z ≤ y + z and 0 ≤ x and 0 ≤ y imply that 0 ≤ x · y for all x, y, z ∈ A .[1] Various extensions of this definition exist that constrain the ring, the partial order, or both. For example, an Archimedean partially ordered ring is a partially ordered ring (A, ≤) where A 's partially ordered additive group is Archimedean.[2] An ordered ring, also called a totally ordered ring, is a partially ordered ring (A, ≤) where ≤ is additionally a total order.[1][2] An l-ring, or lattice-ordered ring, is a partially ordered ring (A, ≤) where ≤ is additionally a lattice order.

187.1 Properties The additive group of a partially ordered ring is always a partially ordered group. The set of non-negative elements of a partially ordered ring (the set of elements x for which 0 ≤ x , also called the positive cone of the ring) is closed under addition and multiplication, i.e., if P is the set of non-negative elements of a partially ordered ring, then P + P ⊆ P , and P · P ⊆ P . Furthermore, P ∩ (−P ) = {0} . The mapping of the compatible partial order on a ring A to the set of its non-negative elements is one-to-one;[1] that is, the compatible partial order uniquely determines the set of non-negative elements, and a set of elements uniquely determines the compatible partial order if one exists. If S is a subset of a ring A, and: 1. 0 ∈ S 2. S ∩ (−S) = {0} 3. S + S ⊆ S 4. S · S ⊆ S then the relation ≤ where x ≤ y iff y − x ∈ S defines a compatible partial order on A (ie. (A, ≤) is a partially ordered ring).[2] In any l-ring, the absolute value |x| of an element x can be defined to be x ∨ (−x) , where x ∨ y denotes the maximal element. For any x and y, 560

187.2. F-RINGS

561

|x · y| ≤ |x| · |y| holds.[3]

187.2 f-rings An f-ring, or Pierce–Birkhoff ring, is a lattice-ordered ring (A, ≤) in which x ∧ y = 0 [4] and 0 ≤ z imply that zx ∧ y = xz ∧ y = 0 for all x, y, z ∈ A . They were first introduced by Garrett Birkhoff and Richard S. Pierce in 1956, in a paper titled “Lattice-ordered rings”, in an attempt to restrict the class of l-rings so as to eliminate a number of pathological examples. For example, Birkhoff and Pierce demonstrated an l-ring with 1 in which 1 is negative, even though being a square.[2] The additional hypothesis required of f-rings eliminates this possibility.

187.2.1

Example

Let X be a Hausdorff space, and C(X) be the space of all continuous, real-valued functions on X. C(X) is an Archimedean f-ring with 1 under the following point-wise operations: [f + g](x) = f (x) + g(x) [f g](x) = f (x) · g(x) [f ∧ g](x) = f (x) ∧ g(x). [2] From an algebraic point of view the rings C(X) are fairly rigid. For example localisations, residue rings or limits of rings of the form C(X) are not of this form in general. A much more flexible class of f-rings containing all rings of continuous functions and resembling many of the properties of these rings, is the class of real closed rings.

187.2.2

Properties

A direct product of f-rings is an f-ring, an l-subring of an f-ring is an f-ring, and an l-homomorphic image of an f-ring is an f-ring.[3] |xy| = |x||y| in an f-ring.[3] The category Arf consists of the Archimedean f-rings with 1 and the l-homomorphisms that preserve the identity.[5] Every ordered ring is an f-ring, so every subdirect union of ordered rings is also an f-ring. Assuming the axiom of choice, a theorem of Birkhoff shows the converse, and that an l-ring is an f-ring if and only if it is l-isomorphic to a subdirect union of ordered rings.[2] Some mathematicians take this to be the definition of an f-ring.[3]

187.3 Formally verified results for commutative ordered rings IsarMathLib, a library for the Isabelle theorem prover, has formal verifications of a few fundamental results on commutative ordered rings. The results are proved in the ring1 context.[6] Suppose (A, ≤) is a commutative ordered ring, and x, y, z ∈ A . Then:

187.4 References [1] Anderson, F. W. “Lattice-ordered rings of quotients”. Canadian Journal of Mathematics: 434–448. doi:10.4153/cjm1965-044-7. [2] Johnson, D. G. (December 1960). “A structure theory for a class of lattice-ordered rings”. Acta Mathematica 104 (3–4): 163–215. doi:10.1007/BF02546389.

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[3] Henriksen, Melvin (1997). “A survey of f-rings and some of their generalizations”. In W. Charles Holland and Jorge Martinez. Ordered Algebraic Structures: Proceedings of the Curaçao Conference Sponsored by the Caribbean Mathematics Foundation, June 23–30, 1995. the Netherlands: Kluwer Academic Publishers. pp. 1–26. ISBN 0-7923-4377-8. [4] ∧ denotes infimum. [5] Hager, Anthony W.; Jorge Martinez (2002). “Functorial rings of quotients—III: The maximum in Archimedean f-rings”. Journal of Pure and Applied Algebra 169: 51–69. doi:10.1016/S0022-4049(01)00060-3. [6] “IsarMathLib” (PDF). Retrieved 2009-03-31.

187.5 Further reading • Birkhoff, G.; R. Pierce (1956). “Lattice-ordered rings”. Anais da Academia Brasileira de Ciências 28: 41–69. • Gillman, Leonard; Jerison, Meyer Rings of continuous functions. Reprint of the 1960 edition. Graduate Texts in Mathematics, No. 43. Springer-Verlag, New York-Heidelberg, 1976. xiii+300 pp

187.6 External links • “Ordered Ring, Partially Ordered Ring”. Encyclopedia of Mathematics. Retrieved 2009-04-03. • “Partially Ordered Ring”. PlanetMath. Retrieved 2009-03-30.

Chapter 188

Perfect field In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds: • Every irreducible polynomial over k has distinct roots. • Every irreducible polynomial over k is separable. • Every finite extension of k is separable. • Every algebraic extension of k is separable. • Either k has characteristic 0, or, when k has characteristic p > 0, every element of k is a pth power. • Either k has characteristic 0, or, when k has characteristic p > 0, the Frobenius endomorphism x→xp is an automorphism of k • The separable closure of k is algebraically closed. • Every reduced commutative k-algebra A is a separable algebra; i.e., A ⊗k F is reduced for every field extension F/k. (see below) Otherwise, k is called imperfect. In particular, all fields of characteristic zero and all finite fields are perfect. Perfect fields are significant because Galois theory over these fields becomes simpler, since the general Galois assumption of field extensions being separable is automatically satisfied over these fields (see third condition above). More generally, a ring of characteristic p (p a prime) is called perfect if the Frobenius endomorphism is an automorphism.[1] (This is equivalent to the above condition “every element of k is a pth power” for integral domains.)

188.1 Examples Examples of perfect fields are: • every field of characteristic zero, e.g. the field of rational numbers or the field of complex numbers; • every finite field, e.g. the field Fp = Z/pZ where p is a prime number; • every algebraically closed field; • the union of a set of perfect fields totally ordered by extension; • fields algebraic over a perfect field. In fact, most fields that appear in practice are perfect. The imperfect case arises mainly in algebraic geometry in characteristic p>0. Every imperfect field is necessarily transcendental over its prime subfield (the minimal subfield), because the latter is perfect. An example of an imperfect field is 563

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CHAPTER 188. PERFECT FIELD

• the field k(X) of all rational functions in an indeterminate X , where k has characteristic p>0 (because X has no p-th root in k(X)).

188.2 Field extension over a perfect field Any finitely generated field extension over a perfect field is separably generated.[2]

188.3 Perfect closure and perfection One of the equivalent conditions says that, in characteristic p, a field adjoined with all pr -th roots (r≥1) is perfect; it −∞ is called the perfect closure of k and usually denoted by k p . The perfect closure can be used in a test for separability. More precisely, a commutative k-algebra A is separable if −∞ and only if A ⊗k k p is reduced.[3] In terms of universal properties, the perfect closure of a ring A of characteristic p is a perfect ring Ap of characteristic p together with a ring homomorphism u : A → Ap such that for any other perfect ring B of characteristic p with a homomorphism v : A → B there is a unique homomorphism f : Ap → B such that v factors through u (i.e. v = fu). The perfect closure always exists; the proof involves “adjoining p-th roots of elements of A", similar to the case of fields.[4] The perfection of a ring A of characteristic p is the dual notion (though this term is sometimes used for the perfect closure). In other words, the perfection R(A) of A is a perfect ring of characteristic p together with a map θ : R(A) → A such that for any perfect ring B of characteristic p equipped with a map φ : B → A, there is a unique map f : B → R(A) such that φ factors through θ (i.e. φ = θf). The perfection of A may be constructed as follows. Consider the projective system

··· → A → A → A → ··· where the transition maps are the Frobenius endomorphism. The inverse limit of this system is R(A) and consists of sequences (x0 , x1 , ... ) of elements of A such that xpi+1 = xi for all i. The map θ : R(A) → A sends (xi) to x0 .[5]

188.4 See also • p-ring • Quasi-finite field

188.5 Notes [1] Serre 1979, Section II.4 [2] Matsumura, Theorem 26.2 [3] Cohn 2003, Theorem 11.6.10 [4] Bourbaki 2003, Section V.5.1.4, page 111 [5] Brinon & Conrad 2009, section 4.2

188.6 References • Bourbaki, Nicolas (2003), Algebra II, Springer, ISBN 978-3-540-00706-7

188.7. EXTERNAL LINKS

565

• Brinon, Olivier; Conrad, Brian (2009), CMI Summer School notes on p-adic Hodge theory (PDF), retrieved 2010-02-05 • Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics 67 (2 ed.), Springer-Verlag, ISBN 978-0-387-90424-5, MR 554237 • Cohn, P.M. (2003), Basic Algebra: Groups, Rings and Fields • Matsumura, H (2003), Commutative ring theory, Translated from the Japanese by M. Reid. Cambridge Studies in Advanced Mathematics 8 (2nd ed.)

188.7 External links • Hazewinkel, Michiel, ed. (2001), “Perfect field”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608010-4

Chapter 189

Perfect ring This article is about perfect rings as introduced by Hyman Bass. For perfect rings of characteristic p generalizing perfect fields, see perfect field. In the area of abstract algebra known as ring theory, a left perfect ring is a type of ring in which all left modules have projective covers. The right case is defined by analogy, and the condition is not left-right symmetric, that is, there exist rings which are perfect on one side but not the other. Perfect rings were introduced in (Bass 1960). A semiperfect ring is a ring over which every finitely generated left module has a projective cover. This property is left-right symmetric.

189.1 Perfect ring 189.1.1

Definitions

The following equivalent definitions of a left perfect ring R are found in (Anderson,Fuller 1992, p.315): • Every left R module has a projective cover. • R/J(R) is semisimple and J(R) is left T-nilpotent (that is, for every infinite sequence of elements of J(R) there is an n such that the product of first n terms are zero), where J(R) is the Jacobson radical of R. • (Bass’ Theorem P) R satisfies the descending chain condition on principal right ideals. (There is no mistake, this condition on right principal ideals is equivalent to the ring being left perfect.) • Every flat left R-module is projective. • R/J(R) is semisimple and every non-zero left R module contains a maximal submodule. • R contains no infinite orthogonal set of idempotents, and every non-zero right R module contains a minimal submodule.

189.1.2

Examples

• Right or left Artinian rings, and semiprimary rings are known to be right-and-left perfect. • The following is an example (due to Bass) of a local ring which is right but not left perfect. Let F be a field, and consider a certain ring of infinite matrices over F. Take the set of infinite matrices with entries indexed by ℕ× ℕ, and which only have finitely many nonzero entries above the diagonal, and denote this set by J. Also take the matrix I with all 1’s on the diagonal, and form the set R = {f · I + j | f ∈ F, j ∈ J} 566

189.2. SEMIPERFECT RING

567

It can be shown that R is a ring with identity, whose Jacobson radical is J. Furthermore R/J is a field, so that R is local, and R is right but not left perfect. (Lam 2001, p.345-346)

189.1.3

Properties

For a left perfect ring R: • From the equivalences above, every left R module has a maximal submodule and a projective cover, and the flat left R modules coincide with the projective left modules. • An analogue of the Baer’s criterion holds for projective modules.

189.2 Semiperfect ring 189.2.1

Definition

Let R be ring. Then R is semiperfect if any of the following equivalent conditions hold: • R/J(R) is semisimple and idempotents lift modulo J(R), where J(R) is the Jacobson radical of R. • R has a complete orthogonal set e1 , ..., en of idempotents with each ei R ei a local ring. • Every simple left (right) R-module has a projective cover. • Every finitely generated left (right) R-module has a projective cover. • The category of finitely generated projective R -modules is Krull-Schmidt.

189.2.2

Examples

Examples of semiperfect rings include: • Left (right) perfect rings. • Local rings. • Left (right) Artinian rings. • Finite dimensional k-algebras.

189.2.3

Properties

Since a ring R is semiperfect iff every simple left R-module has a projective cover, every ring Morita equivalent to a semiperfect ring is also semiperfect.

189.3 References • Anderson, Frank W; Fuller, Kent R (1992), Rings and Categories of Modules, Springer, pp. 312–322, ISBN 0-387-97845-3 • Bass, Hyman (1960), “Finitistic dimension and a homological generalization of semi-primary rings”, Transactions of the American Mathematical Society 95 (3): 466–488, doi:10.2307/1993568, ISSN 0002-9947, JSTOR 1993568, MR 0157984 • Lam, T. Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics 131 (2 ed.), New York: Springer-Verlag, pp. xx+385, ISBN 0-387-95183-0, MR 1838439

Chapter 190

Plane of rotation In geometry, a plane of rotation is an abstract object used to describe or visualise rotations in space. In three dimensions it is an alternative to the axis of rotation, but unlike the axis of rotation it can be used in other dimensions, such as two, four or more dimensions. Mathematically such planes can be described in a number of ways. They can be described in terms of planes and angles of rotation. They can be associated with bivectors from geometric algebra. They are related to the eigenvalues and eigenvectors of a rotation matrix. And in particular dimensions they are related to other algebraic and geometric properties, which can then be generalised to other dimensions. Planes of rotation are not used much in two and three dimensions, as in two dimensions there is only one plane so identifying the plane of rotation is trivial and rarely done, while in three dimensions the axis of rotation serves the same purpose and is the more established approach. The main use for them is in describing more complex rotations in higher dimensions, where they can be used to break down the rotations into simpler parts. This can be done using geometric algebra, with the planes of rotations associated with simple bivectors in the algebra.[1]

190.1 Definitions 190.1.1

Plane

For this article, all planes are planes through the origin, that is they contain the zero vector. Such a plane in ndimensional space is a two-dimensional linear subspace of the space. It is completely specified by any two non-zero and non-parallel vectors that lie in the plane, that is by any two vectors a and b, such that

a ∧ b ̸= 0, where ∧ is the exterior product from exterior algebra or geometric algebra (in three dimensions the cross product can be used). More precisely, the quantity a ∧ b is the bivector associated with the plane specified by a and b, and has magnitude |a| |b| sin φ, where φ is the angle between the vectors; hence the requirement that the vectors be non-zero and non-parallel.[2] If the bivector a ∧ b is written B, then the condition that a point lies on the plane associated with B is simply x ∧ B = 0. [3] This is true in all dimensions, and can be taken as the definition on the plane. In particular, from the properties of the exterior product it is satisfied by both a and b, and so by any vector of the form

c = λa + µb, with λ and μ real numbers. As λ and μ range over all real numbers, c ranges over the whole plane, so this can be taken as another definition of the plane. 568

190.2. TWO DIMENSIONS

190.1.2

569

Plane of rotation

A plane of rotation for a particular rotation is a plane that is mapped to itself by the rotation. The plane is not fixed, but all vectors in the plane are mapped to other vectors in the same plane by the rotation. This transformation of the plane to itself is always a rotation about the origin, through an angle which is the angle of rotation for the plane. Every rotation except for the identity rotation (with matrix the identity matrix) has at least one plane of rotation, and up to ⌊n⌋ 2 planes of rotation, where n is the dimension. The maximum number of planes up to eight dimensions is shown in this table: When a rotation has multiple planes of rotation they are always orthogonal to each other, with only the origin in common. This is a stronger condition than to say the planes are at right angles; it instead means that the planes have no nonzero vectors in common, and that every vector in one plane is orthogonal to every vector in the other plane. This can only happen in four or more dimensions. In two dimensions there is only one plane, while in three dimensions all planes have at least one nonzero vector in common, along their line of intersection.[4] In more than three dimensions planes of rotation are not always unique. For example the negative of the identity matrix in four dimensions (the central inversion), 

−1 0 0  0 −1 0  0 0 −1 0 0 0

 0 0 , 0 −1

describes a rotation in four dimensions in which every plane through the origin is a plane of rotation through an angle π, so any pair of orthogonal planes generates the rotation. But for a general rotation it is at least theoretically possible to identify a unique set of orthogonal planes, in each of which points are rotated through an angle, so the set of planes and angles fully characterise the rotation.[5]

190.2 Two dimensions In two-dimensional space there is only one plane of rotation, the plane of the space itself. In a Cartesian coordinate system it is the Cartesian plane, in complex numbers it is the complex plane. Any rotation therefore is of the whole plane, i.e. of the space, keeping only the origin fixed. It is specified completely by the signed angle of rotation, in the range for example −π to π. So if the angle is θ the rotation in the complex plane is given by Euler’s formula:

eiθ = cos θ + i sin θ, while the rotation in a Cartesian plane is given by the 2×2 rotation matrix: ( cos θ sin θ

) − sin θ [6] . cos θ

190.3 Three dimensions In three-dimensional space there are an infinite number of planes of rotation, only one of which is involved in any given rotation. That is for a general rotation there is precisely one plane which is associated with it or which the rotation takes place in. The only exception is the trivial rotation, corresponding to the identity matrix, in which no rotation takes place. In any rotation in three dimensions there is always a fixed axis, the axis of rotation. The rotation can described by giving this axis, with the angle through which the rotation turns about it; this is the axis angle representation of a

570

CHAPTER 190. PLANE OF ROTATION

Z Y

X

A three-dimensional rotation, with an axis of rotation along the z-axis and a plane of rotation in the xy-plane

rotation. The plane of rotation is the plane orthogonal to this axis, so the axis is a surface normal of the plane. The rotation then rotates this plane through the same angle as it rotates around the axis, that is everything in the plane rotates by the same angle about the origin. One example is shown in the diagram, where the rotation takes place about the z-axis. The plane of rotation is the xy-plane, so everything in that plane it kept in the plane by the rotation. This could be described by a matrix like the following, with the rotation being through an angle θ (about the axis or in the plane):



cos θ  sin θ 0

− sin θ cos θ 0

 0 0. 1

Another example is the Earth’s rotation. The axis of rotation is the line joining the North Pole and South Pole and the plane of rotation is the plane through the equator between the Northern and Southern Hemispheres. Other examples include mechanical devices like a gyroscope or flywheel which store rotational energy in mass usually along the plane of rotation. In any three dimensional rotation the plane of rotation is uniquely defined. Together with the angle of rotation it fully describes the rotation. Or in a continuously rotating object the rotational properties such as the rate of rotation can be described in terms of the plane of rotation. It is perpendicular to, and so is defined by and defines, an axis of rotation, so any description of a rotation in terms of a plane of rotation can be described in terms of an axis of rotation, and vice versa. But unlike the axis of rotation the plane generalises into other, in particular higher, dimensions.[7]

190.4. FOUR DIMENSIONS

571

The Earth showing its axis and plane of rotation, both inclined relative to the plane and perpendicular of Earth’s orbit

190.4 Four dimensions Main article: Rotations in 4-dimensional Euclidean space A general rotation in four-dimensional space has only one fixed point, the origin. Therefore an axis of rotation cannot be used in four dimensions. But planes of rotation can be used, and each non-trivial rotation in four dimensions has one or two planes of rotation.

190.4.1

Simple rotations

A rotation with only one plane of rotation is a simple rotation. In a simple rotation there is a fixed plane, and rotation can be said to take place about this plane, so points as they rotate do not change their distance from this plane. The plane of rotation is orthogonal to this plane, and the rotation can be said to take place in this plane. For example the following matrix fixes the xy-plane: points in that plane and only in that plane are unchanged. The plane of rotation is the zw-plane, points in this plane are rotated through an angle θ. A general point rotates only in the zw-plane, that is it rotates around the xy-plane by changing only its z and w coordinates. 

1 0  0 0

0 1 0 0

0 0 cos θ sin θ

 0 0   − sin θ cos θ

In two and three dimensions all rotations are simple, in that they have only one plane of rotation. Only in four and more dimensions are there rotations that are not simple rotations. In particular in four dimensions there are also double and isoclinic rotations.

572

190.4.2

CHAPTER 190. PLANE OF ROTATION

Double rotations

In a double rotation there are two planes of rotation, no fixed planes, and the only fixed point is the origin. The rotation can be said to take place in both planes of rotation, as points in them are rotated within the planes. These planes are orthogonal, that is they have no vectors in common so every vector in one plane is at right angles to every vector in the other plane. The two rotation planes span four-dimensional space, so every point in the space can be specified by two points, one on each of the planes. A double rotation has two angles of rotation, one for each plane of rotation. The rotation is specified by giving the two planes and two non-zero angles, α and β (if either angle is zero the rotation is simple). Points in the first plane rotate through α, while points in the second plane rotate through β. All other points rotate through an angle between α and β, so in a sense they together determine the amount of rotation. For a general double rotation the planes of rotation and angles are unique, and given a general rotation they can be calculated. For example a rotation of α in the xy-plane and β in the zw-plane is given by the matrix 

cos α  sin α   0 0

190.4.3

− sin α cos α 0 0

0 0 cos β sin β

 0 0  . − sin β  cos β

Isoclinic rotations

A special case of the double rotation is when the angles are equal, that is if α = β ≠ 0. This is called an isoclinic rotation, and it differs from a general double rotation in a number of ways. For example in an isoclinic rotations all non-zero points rotate through the same angle, α. Most importantly the planes of rotation are not uniquely identified. There are instead an infinite number of pairs of orthogonal planes that can be treated as planes of rotation. For example any point can be taken, and the plane it rotates in together with the plane orthogonal to it can be used as two planes of rotation.[8]

190.5 Higher dimensions As already noted the maximum number of planes of rotation in n dimensions is ⌊n⌋

, 2 so the complexity quickly increases with more than four dimensions and categorising rotations as above becomes too complex to be practical, but some observations can be made. Simple rotations can be identified in all dimensions, as rotations with just one plane of rotation. A simple rotation in n dimensions takes place about (that is at a fixed distance from) an (n − 2)-dimensional subspace orthogonal to the plane of rotation. A general rotation is not simple, and has the maximum number of planes of rotation as given above. In the general case the angles of rotations in these planes are distinct and the planes are uniquely defined. If any of the angles are the same then the planes are not unique, as in four dimensions with an isoclinic rotation. In n even dimensions (n = 2, 4, 6...) there are up to n / 2 planes of rotation span the space, so a general rotation rotates all points except the origin which is the only fixed point. In n odd dimensions (n = 3, 5, 7, ...) there are (n − 1) / 2 planes and angles of rotation, the same as the even dimension one lower. These do not span the space, but leave a line which does not rotate – like the axis of rotation in three dimensions, except rotations do not take place about this line but in multiple planes orthogonal to it.[1]

190.6 Mathematical properties The examples given above were chosen to be clear and simple examples of rotations, with planes generally parallel to the coordinate axes in three and four dimensions. But this is not generally the case: planes are not usually parallel

190.6. MATHEMATICAL PROPERTIES

573

A projection of a tesseract with an isoclinic rotation.

to the axes, and the matrices cannot simply be written down. In all dimensions the rotations are fully described by the planes of rotation and their associated angles, so it is useful to be able to determine them, or at least find ways to describe them mathematically.

190.6.1

Reflections

Every simple rotation can be generated by two reflections. Reflections can be specified in n dimensions by giving an (n − 1)-dimensional subspace to reflect in, so a two-dimensional reflection is in a line, a three-dimensional reflection is in a plane, and so on. But this becomes increasingly difficult to apply in higher dimensions, so it is better to use vectors instead, as follows. A reflection in n dimensions is specified by a vector perpendicular to the (n − 1)-dimensional subspace. To generate simple rotations only reflections that fix the origin are needed, so the vector does not have a position, just direction. It does also not matter which way it is facing: it can be replaced with its negative without changing the result. Similarly unit vectors can be used to simplify the calculations. So the reflection in a (n − 1)-dimensional space is given by the unit vector perpendicular to it, m, thus:

574

CHAPTER 190. PLANE OF ROTATION

C

A

C' B'

B

A' θ

θ/2

Two different reflections in two dimensions generating a rotation.

x′ = −mxm where the product is the geometric product from geometric algebra. If x' is reflected in another, distinct, (n − 1)-dimensional space, described by a unit vector n perpendicular to it, the result is

x′′ = −nx′ n = −n(−mxm)n = nmxmn This is a simple rotation in n dimensions, through twice the angle between the subspaces, which is also the angle between the vectors m and n. It can be checked using geometric algebra that this is a rotation, and that it rotates all vectors as expected. The quantity mn is a rotor, and nm is its inverse as

(mn)(nm) = mnnm = mm = 1

190.6. MATHEMATICAL PROPERTIES

575

So the rotation can be written x′′ = RxR−1 where R = mn is the rotor. The plane of rotation is the plane containing m and n, which must be distinct otherwise the reflections are the same and no rotation takes place. As either vector can be replaced by its negative the angle between them can always be acute, or at most π/2. The rotation is through twice the angle between the vectors, up to π or a half-turn. The sense of the rotation is to rotate from m towards n: the geometric product is not commutative so the product nm is the inverse rotation, with sense from n to m. Conversely all simple rotations can be generated this way, with two reflections, by two unit vectors in the plane of rotation separated by half the desired angle of rotation. These can be composed to produce more general rotations, using up to n reflections if the dimension n is even, n − 2 if n is odd, by choosing pairs of reflections given by two vectors in each plane of rotation.[9][10]

190.6.2

Bivectors

Bivectors are quantities from geometric algebra, clifford algebra and the exterior algebra, which generalise the idea of vectors into two dimensions. As vectors are to lines, so are bivectors to planes. So every plane (in any dimension) can be associated with a bivector, and every simple bivector is associated with a plane. This makes them a good fit for describing planes of rotation. Every rotation plane in a rotation has a simple bivector associated with it. This is parallel to the plane and has magnitude equal to the angle of rotation in the plane. These bivectors are summed to produce a single, generally non-simple, bivector for the whole rotation. This can generate a rotor through the exponential map, which can be used to rotate an object. Bivectors are related to rotors through the exponential map (which applied to bivectors generates rotors and rotations using De Moivre’s formula). In particular given any bivector B the rotor associated with it is B

RB = e 2 . This is a simple rotation if the bivector is simple, a more general rotation otherwise. When squared, B

B

RB 2 = e 2 e 2 = eB , it gives a rotor that rotates through twice the angle. If B is simple then this is the same rotation as is generated by two reflections, as the product mn gives a rotation through twice the angle between the vectors. These can be equated, mn = eB , from which it follows that the bivector associated with the plane of rotation containing m and n that rotates m to n is

B = log (mn). This is a simple bivector, associated with the simple rotation described. More general rotations in four or more dimensions are associated with sums of simple bivectors, one for each plane of rotation, calculated as above. Examples include the two rotations in four dimensions given above. The simple rotation in the zw-plane by an angle θ has bivector e34 θ, a simple bivector. The double rotation by α and β in the xy-plane and zw-planes has bivector e12 α + e34 β, the sum of two simple bivectors e12 α and e34 β which are parallel to the two planes of rotation and have magnitudes equal to the angles of rotation. Given a rotor the bivector associated with it can be recovered by taking the logarithm of the rotor, which can then be split into simple bivectors to determine the planes of rotation, although in practice for all but the simplest of cases this may be impractical. But given the simple bivectors geometric algebra is a useful tool for studying planes of rotation using algebra like the above.[1][11]

576

CHAPTER 190. PLANE OF ROTATION

190.6.3

Eigenvalues and eigenplanes

The planes of rotations for a particular rotation using the eigenvalues. Given a general rotation matrix in n dimensions its characteristic equation has either one (in odd dimensions) or zero (in even dimensions) real roots. The other roots are in complex conjugate pairs, exactly ⌊n⌋ 2

,

such pairs. These correspond to the planes of rotation, the eigenplanes of the matrix, which can be calculated using algebraic techniques. In addition arguments of the complex roots are the magnitudes of the bivectors associated with the planes of rotations. The form of the characteristic equation is related to the planes, making it possible to relate its algebraic properties like repeated roots to the bivectors, where repeated bivector magnitudes have particular geometric interpretations.[1][12]

190.7 See also • Charts on SO(3) • Givens rotation • Quaternions • Rotation group SO(3) • Rotations in 4-dimensional Euclidean space

190.8 Notes [1] Lounesto (2001) pp. 222–223 [2] Lounesto (2001) p. 38 [3] Hestenes (1999) p. 48 [4] Lounesto (2001) p. 222 [5] Lounesto (2001) p.87 [6] Lounesto (2001) pp.27–28 [7] Hestenes (1999) pp 280–284 [8] Lounesto (2001) pp. 83–89 [9] Lounesto (2001) p. 57–58 [10] Hestenes (1999) p. 278–280 [11] Dorst, Doran, Lasenby (2002) pp. 79–89 [12] Dorst, Doran, Lasenby (2002) pp. 145–154

190.9 References • Hestenes, David (1999). New Foundations for Classical Mechanics (2nd ed.). Kluwer. ISBN 0-7923-5302-1. • Lounesto, Pertti (2001). Clifford algebras and spinors. Cambridge: Cambridge University Press. ISBN 9780-521-00551-7. • Dorst, Leo; Doran, Chris; Lasenby, Joan (2002). Applications of geometric algebra in computer science and engineering. Birkhäuser. ISBN 0-8176-4267-6.

Chapter 191

Plücker coordinates This article is about the classic case of lines in projective 3-space. For general Plücker coordinates, see Plücker embedding. In geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century, are a way to assign six homogeneous coordinates to each line in projective 3-space, P3 . Because they satisfy a quadratic constraint, they establish a oneto-one correspondence between the 4-dimensional space of lines in P3 and points on a quadric in P5 (projective 5-space). A predecessor and special case of Grassmann coordinates (which describe k-dimensional linear subspaces, or flats, in an n-dimensional Euclidean space), Plücker coordinates arise naturally in geometric algebra. They have proved useful for computer graphics, and also can be extended to coordinates for the screws and wrenches in the theory of kinematics used for robot control.

191.1 Geometric intuition A line L in 3-dimensional Euclidean space is determined by two distinct points that it contains, or by two distinct planes that contain it. Consider the first case, with points x = (x1 ,x2 ,x3 ) and y = (y1 ,y2 ,y3 ). The vector displacement from x to y is nonzero because the points are distinct, and represents the direction of the line. That is, every displacement between points on L is a scalar multiple of d = y−x. If a physical particle of unit mass were to move from x to y, it would have a moment about the origin. The geometric equivalent is a vector whose direction is perpendicular to the plane containing L and the origin, and whose length equals twice the area of the triangle formed by the displacement and the origin. Treating the points as displacements from the origin, the moment is m = x×y, where "×" denotes the vector cross product. For a fixed line, L, the area of the triangle is proportional to the length of the segment between x and y, considered as the base of the triangle; it is not changed by sliding the base along the line, parallel to itself. By definition the moment vector is perpendicular to every displacement along the line, so d•m = 0, where "•" denotes the vector dot product. Although neither d nor m alone is sufficient to determine L, together the pair does so uniquely, up to a common (nonzero) scalar multiple which depends on the distance between x and y. That is, the coordinates (d:m) = (d1 :d2 :d3 :m1 :m2 :m3 ) may be considered homogeneous coordinates for L, in the sense that all pairs (λd:λm), for λ ≠ 0, can be produced by points on L and only L, and any such pair determines a unique line so long as d is not zero and d•m = 0. Furthermore, this approach extends to include points, lines, and a plane “at infinity”, in the sense of projective geometry. Example. Let x = (2,3,7) and y = (2,1,0). Then (d:m) = (0:−2:−7:−7:14:−4). Alternatively, let the equations for points x of two distinct planes containing L be 0 = a + a•x 0 = b + b•x . 577

578

CHAPTER 191. PLÜCKER COORDINATES

Displacement and moment of two points on line

Then their respective planes are perpendicular to vectors a and b, and the direction of L must be perpendicular to both. Hence we may set d = a×b, which is nonzero because a and b are neither zero nor parallel (the planes being distinct and intersecting). If point x satisfies both plane equations, then it also satisfies the linear combination

That is, m = a b − b a is a vector perpendicular to displacements to points on L from the origin; it is, in fact, a moment consistent with the d previously defined from a and b. Example. Let a0 = 2, a = (−1,0,0) and b0 = −7, b = (0,7,−2). Then (d:m) = (0:−2:−7:−7:14:−4). Although the usual algebraic definition tends to obscure the relationship, (d:m) are the Plücker coordinates of L.

191.2 Algebraic definition 191.2.1

Primal coordinates

In a 3-dimensional projective space P3 , let L be a line through distinct points x and y with homogeneous coordinates (x0 :x1 :x2 :x3 ) and (y0 :y1 :y2 :y3 ). The Plücker coordinates pij are defined as follows:

This implies pii = 0 and pij = −pji, reducing the possibilities to only six (4 choose 2) independent quantities. The sixtuple

(p01 : p02 : p03 : p23 : p31 : p12 )

191.2. ALGEBRAIC DEFINITION

579

is uniquely determined by L up to a common nonzero scale factor. Furthermore, not all six components can be zero. Thus the Plücker coordinates of L may be considered as homogeneous coordinates of a point in a 5-dimensional projective space, as suggested by the colon notation. To see these facts, let M be the 4×2 matrix with the point coordinates as columns. 

x0 x1 M = x2 x3

 y0 y1   y2  y3

The Plücker coordinate pij is the determinant of rows i and j of M. Because x and y are distinct points, the columns of M are linearly independent; M has rank 2. Let M′ be a second matrix, with columns x′ and y′ a different pair of distinct points on L. Then the columns of M′ are linear combinations of the columns of M; so for some 2×2 nonsingular matrix Λ, M ′ = M Λ. In particular, rows i and j of M′ and M are related by [

x′i x′j

] [ yi′ x = i yj′ xj

yi yj

][

λ00 λ10

] λ01 . λ11

Therefore, the determinant of the left side 2×2 matrix equals the product of the determinants of the right side 2×2 matrices, the latter of which is a fixed scalar, det Λ. Furthermore, all six 2×2 subdeterminants in M cannot be zero because the rank of M is 2.

191.2.2

Plücker map

Denote the set of all lines (linear images of P1 ) in P3 by G₁,₃. We thus have a map: α : G1,3 → P5 L 7→ Lα , where Lα = (p01 : p02 : p03 : p23 : p31 : p12 ).

191.2.3

Dual coordinates

Alternatively, a line can be described as the intersection of two planes. Let L be a line contained in distinct planes a and b with homogeneous coefficients (a0 :a1 :a2 :a3 ) and (b0 :b1 :b2 :b3 ), respectively. (The first plane equation is ∑k ak xk=0, for example.) The dual Plücker coordinate pij is

Dual coordinates are convenient in some computations, and they are equivalent to primary coordinates: (p01 : p02 : p03 : p23 : p31 : p12 ) = (p23 : p31 : p12 : p01 : p02 : p03 ) Here, equality between the two vectors in homogeneous coordinates means that the numbers on the right side are equal to the numbers on the left side up to some common scaling factor λ . Specifically, let (i,j,k,l) be an even permutation of (0,1,2,3); then pij = λpkl .

580

CHAPTER 191. PLÜCKER COORDINATES

191.2.4

Geometry

To relate back to the geometric intuition, take x0 = 0 as the plane at infinity; thus the coordinates of points not at infinity can be normalized so that x0 = 1. Then M becomes 

1 x1 M = x2 x3

 1 y1  , y2  y3

and setting x = (x1 ,x2 ,x3 ) and y = (y1 ,y2 ,y3 ), we have d = (p01 ,p02 ,p03 ) and m = (p23 ,p31 ,p12 ). Dually, we have d = (p23 ,p31 ,p12 ) and m = (p01 ,p02 ,p03 ).

191.3 Bijection between lines and Klein quadric 191.3.1

Plane equations

If the point z = (z0 :z1 :z2 :z3 ) lies on L, then the columns of  x0 x1  x2 x3

y0 y1 y2 y3

 z0 z1   z2  z3

are linearly dependent, so that the rank of this larger matrix is still 2. This implies that all 3×3 submatrices have determinant zero, generating four (4 choose 3) plane equations, such as

The four possible planes obtained are as follows.

0 = 0 = 0 = 0 =

+ p12 z0 − p31 z0 + p23 z0

− p02 z1 − p03 z1 + p23 z1

+ p01 z2 − p03 z2 + p31 z2

+ p01 z3 + p02 z3 + p12 z3

Using dual coordinates, and letting (a0 :a1 :a2 :a3 ) be the line coefficients, each of these is simply ai = pij , or

0=

3 ∑

pij zi ,

j = 0, . . . , 3.

i=0

Each Plücker coordinate appears in two of the four equations, each time multiplying a different variable; and as at least one of the coordinates is nonzero, we are guaranteed non-vacuous equations for two distinct planes intersecting in L. Thus the Plücker coordinates of a line determine that line uniquely, and the map α is an injection.

191.3.2

Quadratic relation

The image of α is not the complete set of points in P5 ; the Plücker coordinates of a line L satisfy the quadratic Plücker relation

191.4. USES

581

For proof, write this homogeneous polynomial as determinants and use Laplace expansion (in reverse).

Since both 3×3 determinants have duplicate columns, the right hand side is identically zero. Another proof may be done like this: Since vector

is perpendicular to vector

(see above), the scalar product of d and m must be zero! q.e.d.

191.3.3

Point equations

Letting (x0 :x1 :x2 :x3 ) be the point coordinates, four possible points on a line each have coordinates xi = pij, for j = 0…3. Some of these possible points may be inadmissible because all coordinates are zero, but since at least one Plücker coordinate is nonzero, at least two distinct points are guaranteed.

191.3.4

Bijectivity

If (q01 :q02 :q03 :q23 :q31 :q12 ) are the homogeneous coordinates of a point in P5 , without loss of generality assume that q01 is nonzero. Then the matrix 

q01  0 M = −q12 q31

 0 q01   q02  q03

has rank 2, and so its columns are distinct points defining a line L. When the P5 coordinates, qij, satisfy the quadratic Plücker relation, they are the Plücker coordinates of L. To see this, first normalize q01 to 1. Then we immediately have that for the Plücker coordinates computed from M, pij = qij, except for

p23 = −q03 q12 − q02 q31 . But if the qij satisfy the Plücker relation q23 +q02 q31 +q03 q12 = 0, then p23 = q23, completing the set of identities. Consequently, α is a surjection onto the algebraic variety consisting of the set of zeros of the quadratic polynomial

p01 p23 + p02 p31 + p03 p12 . And since α is also an injection, the lines in P3 are thus in bijective correspondence with the points of this quadric in P5 , called the Plücker quadric or Klein quadric.

191.4 Uses Plücker coordinates allow concise solutions to problems of line geometry in 3-dimensional space, especially those involving incidence.

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191.4.1

Line-line crossing

Two lines in P3 are either skew or coplanar, and in the latter case they are either coincident or intersect in a unique point. If pij and p′ij are the Plücker coordinates of two lines, then they are coplanar precisely when d⋅m′+m⋅d′ = 0, as shown by

When the lines are skew, the sign of the result indicates the sense of crossing: positive if a right-handed screw takes L into L′, else negative. The quadratic Plücker relation essentially states that a line is coplanar with itself.

191.4.2

Line-line join

In the event that two lines are coplanar but not parallel, their common plane has equation 0 = (m•d′)x0 + (d×d′)•x , where x = (x1 ,x2 ,x3 ). The slightest perturbation will destroy the existence of a common plane, and near-parallelism of the lines will cause numeric difficulties in finding such a plane even if it does exist.

191.4.3

Line-line meet

Dually, two coplanar lines, neither of which contains the origin, have common point (x0 : x) = (d•m′:m×m′) . To handle lines not meeting this restriction, see the references.

191.4.4

Plane-line meet

Given a plane with equation

0 = a0 x0 + a1 x1 + a2 x2 + a3 x3 , or more concisely 0 = a0 x0 +a•x; and given a line not in it with Plücker coordinates (d:m), then their point of intersection is (x0 : x) = (a•d : a×m − a0 d) . The point coordinates, (x0 :x1 :x2 :x3 ), can also be expressed in terms of Plücker coordinates as

xi =

∑

aj pij ,

i = 0 . . . 3.

j̸=i

191.4.5

Point-line join

Dually, given a point (y0 :y) and a line not containing it, their common plane has equation 0 = (y•m) x0 + (y×d−y0 m)•x .

191.5. SEE ALSO

583

The plane coordinates, (a0 :a1 :a2 :a3 ), can also be expressed in terms of dual Plücker coordinates as ai =

∑

yj pij ,

i = 0 . . . 3.

j̸=i

191.4.6

Line families

Because the Klein quadric is in P5 , it contains linear subspaces of dimensions one and two (but no higher). These correspond to one- and two-parameter families of lines in P3 . For example, suppose L and L′ are distinct lines in P3 determined by points x, y and x′, y′, respectively. Linear combinations of their determining points give linear combinations of their Plücker coordinates, generating a oneparameter family of lines containing L and L′. This corresponds to a one-dimensional linear subspace belonging to the Klein quadric. Lines in plane If three distinct and non-parallel lines are coplanar; their linear combinations generate a two-parameter family of lines, all the lines in the plane. This corresponds to a two-dimensional linear subspace belonging to the Klein quadric. Lines through point If three distinct and non-coplanar lines intersect in a point, their linear combinations generate a two-parameter family of lines, all the lines through the point. This also corresponds to a two-dimensional linear subspace belonging to the Klein quadric. Ruled surface A ruled surface is a family of lines that is not necessarily linear. It corresponds to a curve on the Klein quadric. For example, a hyperboloid of one sheet is a quadric surface in P3 ruled by two different families of lines, one line of each passing through each point of the surface; each family corresponds under the Plücker map to a conic section within the Klein quadric in P5 .

191.4.7

Line geometry

During the nineteenth century, line geometry was studied intensively. In terms of the bijection given above, this is a description of the intrinsic geometry of the Klein quadric. Ray tracing Line geometry is extensively used in ray tracing application where the geometry and intersections of rays need to be calculated in 3D. An implementation is described in Introduction to Pluecker Coordinates written for the Ray Tracing forum by Thouis Jones.

191.5 See also • Flat projective plane

191.6 References • Hodge, W. V. D.; D. Pedoe (1994) [1947]. Methods of Algebraic Geometry, Volume I (Book II). Cambridge University Press. ISBN 978-0-521-46900-5.

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• Behnke, H.; F. Bachmann, K. Fladt, H. Kunle (eds.) (1984). Fundamentals of Mathematics, Volume II: Geometry. trans. S. H. Gould. MIT Press. ISBN 978-0-262-52094-2. From the German: Grundzüge der Mathematik, Band II: Geometrie. Vandenhoeck & Ruprecht. • Guilfoyle, B.; W. Klingenberg (2004). “On the space of oriented affine lines in R^3” (PDF). Archiv der Mathematik (Birkhäuser) 82 (1): 81–84. ISSN 0003-889X. • Kuptsov, L.P. (2001), “P/p072890”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Mason, Matthew T.; J. Kenneth Salisbury (1985). Robot Hands and the Mechanics of Manipulation. MIT Press. ISBN 978-0-262-13205-3. • Hohmeyer, M.; S. Teller (1999). “Determining the Lines Through Four Lines” (PDF). Journal of Graphics Tools (A K Peters) 4 (3): 11–22. ISSN 1086-7651. • Shafarevich, I. R.; A. O. Remizov (2012). Linear Algebra and Geometry. Springer. ISBN 978-3-642-30993-9.

Chapter 192

Poisson ring In mathematics, a Poisson ring is a commutative ring on which an anticommutative and distributive binary operation [·, ·] satisfying the Jacobi identity and the product rule is defined. Such an operation is then known as the Poisson bracket of the Poisson ring. Many important operations and results of symplectic geometry and Hamiltonian mechanics may be formulated in terms of the Poisson bracket and, hence, apply to Poisson algebras as well. This observation is important in studying the classical limit of quantum mechanics—the non-commutative algebra of operators on a Hilbert space has the Poisson algebra of functions on a symplectic manifold as a singular limit, and properties of the non-commutative algebra pass over to corresponding properties of the Poisson algebra.

192.1 Definition The Poisson bracket must satisfy the identities • [f, g] = −[g, f ] (skew symmetry) • [f + g, h] = [f, h] + [g, h] (distributivity) • [f g, h] = f [g, h] + [f, h]g (derivation) • [f, [g, h]] + [g, [h, f ]] + [h, [f, g]] = 0 (Jacobi identity) for all f, g, h in the ring. A Poisson algebra is a Poisson ring that is also an algebra over a field. In this case, add the extra requirement

[sf, g] = s[f, g] for all scalars s. For each g in a Poisson ring A, the operation adg defined as adg (f ) = [f, g] is a derivation. If the set {adg |g ∈ A} generates the set of derivations of A, then A is said to be non-degenerate. If a non-degenerate Poisson ring is isomorphic as a commutative ring to the algebra of smooth functions on a manifold M, then M must be a symplectic manifold and [·, ·] is the Poisson bracket defined by the symplectic form.

192.2 References • If the algebra of functions on a manifold is a Poisson ring then the manifold is symplectic at PlanetMath.org. This article incorporates material from Poisson Ring on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. 585

Chapter 193

Polynomial identity ring In mathematics, in the subfield of ring theory, a ring R is a polynomial identity ring if there is, for some N > 0, an element P other than 0 of the free algebra, Z, over the ring of integers in N variables X1 , X2 , ..., XN such that for all N-tuples r1 , r2 , ..., rN taken from R it happens that

P (r1 , r2 , . . . , rN ) = 0. Strictly the Xi here are “non-commuting indeterminates”, and so “polynomial identity” is a slight abuse of language, since “polynomial” here stands for what is usually called a “non-commutative polynomial”. The abbreviation PI-ring is common. More generally, the free algebra over any ring S may be used, and gives the concept of PI-algebra. If the degree of the polynomial P is defined in the usual way, the polynomial P is called monic if at least one of its terms of highest degree has coefficient equal to 1. Every commutative ring is a PI-ring, satisfying the polynomial identity XY - YX = 0. Therefore PI-rings are usually taken as close generalizations of commutative rings. If the ring has characteristic p different from zero then it satisfies the polynomial identity pX = 0. To exclude such examples, sometimes it is defined that PI-rings must satisfy a monic polynomial identity.[1]

193.1 Examples • For example if R is a commutative ring it is a PI-ring: this is true with

P (X1 , X2 ) = X1 X2 − X2 X1 = 0 • The ring of 2 by 2 matrices over a commutative ring satisfies the Hall identity

(xy − yx)2 z = z(xy − yx)2 This identity was used by M. Hall (1943), but was found earlier by Wagner (1937). • A major role is played in the theory by the standard identity sN, of length N, which generalises the example given for commutative rings (N = 2). It derives from the Leibniz formula for determinants

det(A) =

∑ σ∈SN

sgn(σ)

N ∏

ai,σ(i)

i=1

586

193.2. PROPERTIES

587

by replacing each product in the summand by the product of the Xᵢ in the order given by the permutation σ. In other words each of the N! orders is summed, and the coefficient is 1 or −1 according to the signature.

∑

sN (X1 , . . . , XN ) =

sgn(σ)Xσ(1) · · · Xσ(N ) = 0

σ∈SN

The m×m matrix ring over any commutative ring satisfies a standard identity: the Amitsur–Levitzki theorem states that it satisfies s₂m. The degree of this identity is optimal since the matrix ring can not satisfy any monic polynomial of degree less than 2m. • Given a field k of characteristic zero, take R to be the exterior algebra over a countably infinite-dimensional vector space with basis e1 , e2 , e3 , ... Then R is generated by the elements of this basis and eiej = −ejei. This ring does not satisfy sN for any N and therefore can not be embedded in any matrix ring. In fact sN(e1 ,e2 ,...,eN) = N!e1 e2 ...eN ≠ 0. On the other hand it is a PI-ring since it satisfies [[x, y], z] := xyz − yxz − zxy + zyx = 0. It is enough to check this for monomials in the e's. Now, a monomial of even degree commutes with every element. Therefore if either x or y is a monomial of even degree [x, y] := xy − yx = 0. If both are of odd degree then [x, y] = xy − yx = 2xy has even degree and therefore commutes with z, i.e. [[x, y], z] = 0.

193.2 Properties • Any subring or homomorphic image of a PI-ring is a PI-ring. • A finite direct product of PI-rings is a PI-ring. • A direct product of PI-rings, satisfying the same identity, is a PI-ring. • It can always be assumed that the identity that the PI-ring satisfies is multilinear. • If a ring is finitely generated by n elements as a module over its center then it satisfies every alternating multilinear polynomial of degree larger than n. In particular it satisfies sN for N > n and therefore it is a PI-ring. • If R and S are PI-rings then their tensor product over the integers, R ⊗Z S , is also a PI-ring. • If R is a PI-ring, then so is the ring of n×n-matrices with coefficients in R.

193.3 PI-rings as generalizations of commutative rings Among noncommutative rings, PI-rings satisfy the Köthe conjecture. Affine PI-algebras over a field satisfy the Kurosh conjecture, the Nullstellensatz and the catenary property for prime ideals. If R is a PI-ring and K is a subring of its center such that R is integal over K then the going up and going down properties for prime ideals of R and K are satisfied. Also the lying over property (If p is a prime ideal of K then there is a prime ideal P of R such that p is minimal over P ∩ K ) and the incomparability property (If P and Q are prime ideals of R and P ⊂ Q then P ∩ K ⊂ Q ∩ K ) are satisfied.

193.4 The set of identities a PI-ring satisfies If F := Z is the free algebra in N variables and R is a PI-ring satisfying the polynomial P in N variables, then P is in the kernel of any homomorphism

588

CHAPTER 193. POLYNOMIAL IDENTITY RING τ :F → R.

An ideal I of F is called T-ideal if f (I) ⊂ I for every endomorphism f of F. Given a PI-ring, R, the set of all polynomial identities it satisfies is an ideal but even more it is a T-ideal. Conversely, if I is a T-ideal of F then F/I is a PI-ring satisfying all identities in I. It is assumed that I contains monic polynomials when PI-rings are required to satisfy monic polynomial identities.

193.5 See also • Posner’s theorem • Central polynomial

193.6 References [1] J.C. McConnell, J.C. Robson, Noncommutative Noetherian Rings, Graduate studies in Mathematics, Vol 30

• Latyshev, V.N. (2001), “PI-algebra”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Formanek, E. (2001), “Amitsur–Levitzki theorem”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Polynomial identities in ring theory, Louis Halle Rowen, Academic Press, 1980, ISBN 978-0-12-599850-5 • Polynomial identity rings, Vesselin S. Drensky, Edward Formanek, Birkhäuser, 2004, ISBN 978-3-7643-71265 • Polynomial identities and asymptotic methods, A. Giambruno, Mikhail Zaicev, AMS Bookstore, 2005, ISBN 978-0-8218-3829-7 • Computational aspects of polynomial identities, Alexei Kanel-Belov, Louis Halle Rowen, A K Peters Ltd., 2005, ISBN 978-1-56881-163-5

193.7 Further reading • Formanek, Edward (1991). The polynomial identities and invariants of n×n matrices. Regional Conference Series in Mathematics 78. Providence, RI: American Mathematical Society. ISBN 0-8218-0730-7. Zbl 0714.16001. • Kanel-Belov, Alexei; Rowen, Louis Halle (2005). Computational aspects of polynomial identities. Research Notes in Mathematics 9. Wellesley, MA: A K Peters. ISBN 1-56881-163-2. Zbl 1076.16018.

193.8 External links • Polynomial identity algebra at PlanetMath.org. • Standard Identity at PlanetMath.org. • T-ideal at PlanetMath.org.

Chapter 194

Polynomial ring In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of splitting fields, and to the understanding of a linear operator. Many important conjectures involving polynomial rings, such as Serre’s problem, have influenced the study of other rings, and have influenced even the definition of other rings, such as group rings and rings of formal power series. A closely related notion is that of the ring of polynomial functions on a vector space.

194.1 The polynomial ring K[X] 194.1.1

Definition

The polynomial ring, K[X], in X over a field K is defined[1] as the set of expressions, called polynomials in X, of the form

p = p0 + p1 X + p2 X 2 + · · · + pm−1 X m−1 + pm X m , where p0 , p1 ,…, p , the coefficients of p, are elements of K, and X, X 2 , are formal symbols (“the powers of X"). By convention, X 0 = 1, X 1 = X, and the product of the powers of X is defined by the familiar formula

X k X l = X k+l , where k and l are any natural numbers. The symbol X is called an indeterminate[2] or variable.[3] Two polynomials are defined to be equal if and only if the corresponding coefficients for each power of X are equal, however terms with zero coefficient, 0X k , may be added or omitted. This terminology is suggested by real or complex polynomial functions. However, in general, X and its powers, X k , are treated as formal symbols, not as elements of the field K or functions over it. One can think of the ring K[X] as arising from K by adding one new element X that is external to K and requiring that X commute with all elements of K. Polynomials in X are added and multiplied according to the ordinary rules for manipulating algebraic expressions, creating the structure of a ring. Specifically, if

p = p0 + p1 X + p2 X 2 + · · · + pm X m , and 589

590

CHAPTER 194. POLYNOMIAL RING

q = q0 + q1 X + q2 X 2 + · · · + qn X n , then

p + q = r0 + r1 X + r2 X 2 + · · · + rk X k , and

pq = s0 + s1 X + s2 X 2 + · · · + sl X l , where

ri = pi + qi and

si = p0 qi + p1 qi−1 + · · · + pi q0 . If necessary, the polynomials p and q are extended by adding “dummy terms” with zero coefficients, so that the expressions for ri and si are always defined. A more rigorous, but less intuitive treatment[4] defines a polynomial as an infinite tuple, or ordered sequence of elements of K, (p0 , p1 , p2 , … ) having the property that only a finite number of the elements are nonzero, or equivalently, a sequence for which there is some m so that pn = 0 for n>m. In this case, the expression

p0 + p1 X + p2 X 2 + · · · + pm X m is considered an alternate notation for the sequence (p0 , p1 , p2 , … pm, 0, 0, …). More generally, the field K can be replaced by any commutative ring R when taking the same construction as above, giving rise to the polynomial ring over R, which is denoted R[X].

194.1.2

Degree of a polynomial

The degree of a polynomial p, written deg(p) is the largest k such that the coefficient of X k is not zero.[5] In this case the coefficient pk is called the leading coefficient.[6] In the special case of zero polynomial, all of whose coefficients are zero, the degree has been variously left undefined,[7] defined to be −1,[8] or defined to be a special symbol −∞.[9] If K is a field, or more generally an integral domain, then from the definition of multiplication,[10]

deg(pq) = deg(p) + deg(q). It follows immediately that if K is an integral domain then so is K[X].[11]

194.1.3

Properties of K[X]

Factorization in K[X] The next property of the polynomial ring is much deeper. Already Euclid noted that every positive integer can be uniquely factored into a product of primes — this statement is now called the fundamental theorem of arithmetic. The proof is based on Euclid’s algorithm for finding the greatest common divisor of natural numbers. At each step

194.1. THE POLYNOMIAL RING K[X]

591

of this algorithm, a pair (a, b), a > b, of natural numbers is replaced by a new pair (b, r), where r is the remainder from the division of a by b, and the new numbers are smaller. Gauss remarked that the procedure of division with the remainder can also be defined for polynomials: given two polynomials p and q, where q ≠ 0, one can write

p = uq + r, where the quotient u and the remainder r are polynomials, the degree of r is less than the degree of q, and a decomposition with these properties is unique. The quotient and the remainder are found using the polynomial long division. The degree of the polynomial now plays a role similar to the absolute value of an integer: it is strictly less in the remainder r than it is in q, and when repeating this step such decrease cannot go on indefinitely. Therefore eventually some division will be exact, at which point the last non-zero remainder is the greatest common divisor of the initial two polynomials. Using the existence of greatest common divisors, Gauss was able to simultaneously rigorously prove the fundamental theorem of arithmetic for integers and its generalization to polynomials. In fact there exist other commutative rings than Z and K[X] that similarly admit an analogue of the Euclidean algorithm; all such rings are called Euclidean rings. Rings for which there exists unique (in an appropriate sense) factorization of nonzero elements into irreducible factors are called unique factorization domains or factorial rings; the given construction shows that all Euclidean rings, and in particular Z and K[X], are unique factorization domains. Another corollary of the polynomial division with the remainder is the fact that every proper ideal I of K[X] is principal, i.e. I consists of the multiples of a single polynomial f. Thus the polynomial ring K[X] is a principal ideal domain, and for the same reason every Euclidean domain is a principal ideal domain. Also every principal ideal domain is a unique-factorization domain. These deductions make essential use of the fact that the polynomial coefficients lie in a field, namely in the polynomial division step, which requires the leading coefficient of q, which is only known to be non-zero, to have an inverse. If R is an integral domain that is not a field then R[X] is neither a Euclidean domain nor a principal ideal domain; however it could still be a unique factorization domain (and will be so if and only it R itself is a unique factorization domain, for instance if it is Z or another polynomial ring). Quotient ring of K[X] The ring K[X] of polynomials over K is obtained from K by adjoining one element, X. It turns out that any commutative ring L containing K and generated as a ring by a single element in addition to K can be described using K[X]. In particular, this applies to finite field extensions of K. Suppose that a commutative ring L contains K and there exists an element θ of L such that the ring L is generated by θ over K. Thus any element of L is a linear combination of powers of θ with coefficients in K. Then there is a unique ring homomorphism φ from K[X] into L which does not affect the elements of K itself (it is the identity map on K) and maps each power of X to the same power of θ. Its effect on the general polynomial amounts to “replacing X with θ":

φ(am X m + am−1 X m−1 + · · · + a1 X + a0 ) = am θm + am−1 θm−1 + · · · + a1 θ + a0 . By the assumption, any element of L appears as the right hand side of the last expression for suitable m and elements a0 , …, am of K. Therefore, φ is surjective and L is a homomorphic image of K[X]. More formally, let Ker φ be the kernel of φ. It is an ideal of K[X] and by the first isomorphism theorem for rings, L is isomorphic to the quotient of the polynomial ring K[X] by the ideal Ker φ. Since the polynomial ring is a principal ideal domain, this ideal is principal: there exists a polynomial p∈K[X] such that

L ≃ K[X]/(p). A particularly important application is to the case when the larger ring L is a field. Then the polynomial p must be irreducible. Conversely, the primitive element theorem states that any finite separable field extension L/K can be generated by a single element θ∈L and the preceding theory then gives a concrete description of the field L as the quotient of the polynomial ring K[X] by a principal ideal generated by an irreducible polynomial p. As an illustration, the field C of complex numbers is an extension of the field R of real numbers generated by a single element i such that i2 + 1 = 0. Accordingly, the polynomial X2 + 1 is irreducible over R and

592

CHAPTER 194. POLYNOMIAL RING

C ≃ R[X]/(X 2 + 1). More generally, given a (not necessarily commutative) ring A containing K and an element a of A that commutes with all elements of K, there is a unique ring homomorphism from the polynomial ring K[X] to A that maps X to a:

ϕ : K[X] → A,

ϕ(X) = a.

This homomorphism is given by the same formula as before, but it is not surjective in general. The existence and uniqueness of such a homomorphism φ expresses a certain universal property of the ring of polynomials in one variable and explains ubiquity of polynomial rings in various questions and constructions of ring theory and commutative algebra.

194.1.4

Modules

The structure theorem for finitely generated modules over a principal ideal domain applies over K[X]. This means that every finitely generated module over K[X] may be decomposed into a direct sum of a free module and finitely many modules of the form K[X]/⟨P k ⟩ , where P is an irreducible polynomial over K and k a positive integer.

194.2 Polynomial evaluation Let K be a field or, more generally, a commutative ring, and R a ring containing K. For any polynomial P in K[X] and any element a in R, the substitution of X by a in P defines an element of R, which is denoted P(a). This element is obtained by, after the substitution, carrying on, in R, the operations indicated by the expression of the polynomial. This computation is called the evaluation of P at a. For example, if we have

P = X 2 − 1, we have

P (3) = 32 − 1 = 8, P (X 2 + 1) = (X 2 + 1)2 − 1 = X 4 + 2X 2 (in the first example R = K, and in the second one R = K[X]). Substituting X by itself results in

P = P (X), explaining why the sentences "Let P be a polynomial" and "Let P(X) be a polynomial" are equivalent. For every a in R, the map P 7→ P (a) defines a ring homomorphism from K[X] into R. The polynomial function defined by a polynomial P is the function from K into K that is defined by x 7→ P (x). If K is an infinite field, two different polynomials define different polynomial functions, but this property is false for finite fields. For example, if K is a field with q elements, then the polynomials 0 and Xq -X both define the zero function.

194.3 The polynomial ring in several variables 194.3.1

Polynomials

A polynomial in n variables X1 , …, Xn with coefficients in a field K is defined analogously to a polynomial in one variable, but the notation is more cumbersome. For any multi-index α = (α1 , …, αn), where each αi is a non-negative integer, let

194.3. THE POLYNOMIAL RING IN SEVERAL VARIABLES

n ∏

Xα =

593

Xiαi = X1α1 . . . Xnαn .

i=1

The product Xα is called the monomial of multidegree α. A polynomial is a finite linear combination of monomials with coefficients in K

p=

∑

pα X α ,

α

where pα = pα1 ,...,αn ∈ K, and only finitely many coefficients pα are different from 0. The degree of a monomial Xα , frequently denoted |α|, is defined as

|α| =

n ∑

αi ,

i=1

and the degree of a polynomial p is the largest degree of a monomial occurring with non-zero coefficient in the expansion of p.

194.3.2

The polynomial ring

Polynomials in n variables with coefficients in K form a commutative ring denoted K[X1 ,…, Xn], or sometimes K[X], where X is a symbol representing the full set of variables, X = (X1 ,…, Xn), and called the polynomial ring in n variables. The polynomial ring in n variables can be obtained by repeated application of K[X] (the order by which is irrelevant). For example, K[X1 , X2 ] is isomorphic to K[X1 ][X2 ]. This ring plays fundamental role in algebraic geometry. Many results in commutative and homological algebra originated in the study of its ideals and modules over this ring. A polynomial ring with coefficients in Z is the free commutative ring over its set of variables.

194.3.3

Hilbert’s Nullstellensatz

Main article: Hilbert’s Nullstellensatz A group of fundamental results concerning the relation between ideals of the polynomial ring K[X1 ,…, Xn] and algebraic subsets of K n originating with David Hilbert is known under the name Nullstellensatz (literally: “zerolocus theorem”). • (Weak form, algebraically closed field of coefficients). Let K be an algebraically closed field. Then every maximal ideal m of K[X1 ,…, Xn] has the form

m = (X1 − a1 , . . . , Xn − an ),

a = (a1 , . . . , an ) ∈ K n .

• (Weak form, any field of coefficients). Let k be a field, K be an algebraically closed field extension of k, and I be an ideal in the polynomial ring k[X1 ,…, Xn]. Then I contains 1 if and only if the polynomials in I do not have any common zero in K n . • (Strong form). Let k be a field, K be an algebraically closed field extension of k, I be an ideal in the polynomial ring k[X1 ,…, Xn],and V(I) be the algebraic subset of K n defined by I. Suppose that f is a polynomial which vanishes at all points of V(I). Then some power of f belongs to the ideal I:

594

CHAPTER 194. POLYNOMIAL RING

f m ∈ I, some for m ∈ N. Using the notion of the radical of an ideal, the conclusion says that f belongs to the radical of I. As a corollary of this form of Nullstellensatz, there is a bijective correspondence between the radical ideals of K[X1 ,…, Xn] for an algebraically closed field K and the algebraic subsets of the n-dimensional affine space K n . It arises from the map

I 7→ V (I),

I ⊂ K[X1 , . . . , Xn ],

V (I) ⊂ K n .

The prime ideals of the polynomial ring correspond to irreducible subvarieties of K n .

194.4 Properties of the ring extension R ⊂ R[X] One of the basic techniques in commutative algebra is to relate properties of a ring with properties of its subrings. The notation R ⊂ S indicates that a ring R is a subring of a ring S. In this case S is called an overring of R and one speaks of a ring extension. This works particularly well for polynomial rings and allows one to establish many important properties of the ring of polynomials in several variables over a field, K[X1 ,…, Xn], by induction in n.

194.4.1

Summary of the results

In the following properties, R is a commutative ring and S = R[X1 ,…, Xn] is the ring of polynomials in n variables over R. The ring extension R ⊂ S can be built from R in n steps, by successively adjoining X1 ,…, Xn. Thus to establish each of the properties below, it is sufficient to consider the case n = 1. • If R is an integral domain then the same holds for S. • If R is a unique factorization domain then the same holds for S. The proof is based on the Gauss lemma. • Hilbert’s basis theorem: If R is a Noetherian ring, then the same holds for S. • Suppose that R is a Noetherian ring of finite global dimension. Then

gl dim R[X1 , . . . , Xn ] = gl dim R + n. An analogous result holds for Krull dimension.

194.5 Generalizations Polynomial rings have been generalized in a great many ways, including polynomial rings with generalized exponents, power series rings, noncommutative polynomial rings, and skew-polynomial rings.

194.5.1

Infinitely many variables

The possibility to allow an infinite set of indeterminates is not really a generalization, as the ordinary notion of polynomial ring allows for it. It is then still true that each monomial involves only a finite number of indeterminates (so that its degree remains finite), and that each polynomial is a linear combination of monomials, which by definition involves only finitely many of them. This explains why such polynomial rings are relatively seldom considered:

194.5. GENERALIZATIONS

595

each individual polynomial involves only finitely many indeterminates, and even any finite computation involving polynomials remains inside some subring of polynomials in finitely many indeterminates. In the case of infinitely many indeterminates, one can consider a ring strictly larger than the polynomial ring but smaller than the power series ring, by taking the subring of the latter formed by power series whose monomials have a bounded degree. Its elements still have a finite degree and are therefore are somewhat like polynomials, but it is possible for instance to take the sum of all indeterminates, which is not a polynomial. A ring of this kind plays a role in constructing the ring of symmetric functions.

194.5.2

Generalized exponents

Main article: Monoid ring A simple generalization only changes the set from which the exponents on the variable are drawn. The formulas for addition and multiplication make sense as long as one can add exponents: Xi ·Xj = Xi+j . A set for which addition makes sense (is closed and associative) is called a monoid. The set of functions from a monoid N to a ring R which are nonzero at only finitely many places can be given the structure of a ring known as R[N], the monoid ring of N with coefficients in R. The addition is defined component-wise, so that if c = a+b, then cn = an + bn for every n in N. The multiplication is defined as the Cauchy product, so that if c = a·b, then for each n in N, cn is the sum of all aibj where i, j range over all pairs of elements of N which sum to n. When N is commutative, it is convenient to denote the function a in R[N] as the formal sum: ∑

an X n

n∈N

and then the formulas for addition and multiplication are the familiar: (

∑

) an X

n

( +

n∈N

∑

) bn X

n

=

n∈N

∑

(an + bn )X n

n∈N

and (

∑

) ( an X n

n∈N

·

∑

) bn X n

=

n∈N

∑ n∈N

 

∑

 ai bj  X n

i+j=n

where the latter sum is taken over all i, j in N that sum to n. Some authors such as (Lang 2002, II,§3) go so far as to take this monoid definition as the starting point, and regular single variable polynomials are the special case where N is the monoid of non-negative integers. Polynomials in several variables simply take N to be the direct product of several copies of the monoid of non-negative integers. Several interesting examples of rings and groups are formed by taking N to be the additive monoid of non-negative rational numbers, (Osbourne 2000, §4.4).

194.5.3

Power series

Main article: Formal power series Power series generalize the choice of exponent in a different direction by allowing infinitely many nonzero terms. This requires various hypotheses on the monoid N used for the exponents, to ensure that the sums in the Cauchy product are finite sums. Alternatively, a topology can be placed on the ring, and then one restricts to convergent infinite sums. For the standard choice of N, the non-negative integers, there is no trouble, and the ring of formal power series is defined as the set of functions from N to a ring R with addition component-wise, and multiplication given by the Cauchy product. The ring of power series can be seen as the completion of the polynomial ring.

596

194.5.4

CHAPTER 194. POLYNOMIAL RING

Noncommutative polynomial rings

Main article: Free algebra For polynomial rings of more than one variable, the products X·Y and Y·X are simply defined to be equal. A more general notion of polynomial ring is obtained when the distinction between these two formal products is maintained. Formally, the polynomial ring in n noncommuting variables with coefficients in the ring R is the monoid ring R[N], where the monoid N is the free monoid on n letters, also known as the set of all strings over an alphabet of n symbols, with multiplication given by concatenation. Neither the coefficients nor the variables need commute amongst themselves, but the coefficients and variables commute with each other. Just as the polynomial ring in n variables with coefficients in the commutative ring R is the free commutative R-algebra of rank n, the noncommutative polynomial ring in n variables with coefficients in the commutative ring R is the free associative, unital R-algebra on n generators, which is noncommutative when n > 1.

194.5.5

Differential and skew-polynomial rings

Main article: Ore extension Other generalizations of polynomials are differential and skew-polynomial rings. A differential polynomial ring is a ring of differential operators formed from a ring R and a derivation δ of R into R. This derivation operates on R, and will be denoted X, when viewed as an operator. The elements of R also operate on R by multiplication. The composition of operators is denoted as the usual multiplication. It follows that the relation δ(ab) = aδ(b) + δ(a)b may be rewritten

X · a = a · X + δ(a). This relation may be extended to define a skew multiplication between two polynomials in X with coefficients in R, which make them a non-commutative ring. The standard example, called a Weyl algebra, takes R to be a (usual) polynomial ring k[Y], and δ to be the standard ∂ polynomial derivative ∂Y . Taking a =Y in the above relation, one gets the canonical commutation relation, X·Y − Y·X = 1. Extending this relation by associativity and distributivity allows to construct explicitly the Weyl algebra.(Lam 2001, §1,ex1.9). The skew-polynomial ring is defined similarly for a ring R and a ring endomorphism f of R, by extending the multiplication from the relation X·r = f(r)·X to produce an associative multiplication that distributes over the standard addition. More generally, given a homomorphism F from the monoid N of the positive integers into the endomorphism ring of R, the formula Xn ·r = F(n)(r)·Xn allows to construct a skew-polynomial ring.(Lam 2001, §1,ex 1.11) Skew polynomial rings are closely related to crossed product algebras.

194.6 See also • Additive polynomial • Laurent polynomial

194.7 References [1] Following Herstein p. 153 [2] Herstein, Hall p. 73 [3] Lang p. 97 [4] Following Hall p.72-73

194.7. REFERENCES

597

[5] Herstein p. 154 [6] Lang p.100 [7] Anton, Howard; Bivens, Irl C.; Davis, Stephen (2012), Calculus Single Variable, John Wiley & Sons, p. 31, ISBN 9780470647707. [8] Sendra, J. Rafael; Winkler, Franz; Pérez-Diaz, Sonia (2007), Rational Algebraic Curves: A Computer Algebra Approach, Algorithms and Computation in Mathematics 22, Springer, p. 250, ISBN 9783540737247. [9] Eves, Howard Whitley (1980), Elementary Matrix Theory, Dover, p. 183, ISBN 9780486150277. [10] Herstein p.155, 162 [11] Herstein p.162

• Hall, F. M. (1969). “Section 3.6”. An Introduction to Abstract Algebra 2. Cambridge University Press. ISBN 0521084849. • Herstein, I. N. (1975). “Section 3.9”. Topics in Algebra. Wiley. ISBN 0471010901. • Lam, Tsit-Yuen (2001), A First Course in Noncommutative Rings, Berlin, New York: Springer-Verlag, ISBN 978-0-387-95325-0 • Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: SpringerVerlag, ISBN 978-0-387-95385-4, MR 1878556 • Osborne, M. Scott (2000), Basic homological algebra, Graduate Texts in Mathematics 196, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98934-1, MR 1757274

Chapter 195

Posner’s theorem In algebra, Posner’s theorem states that given a prime polynomial identity algebra A with center Z, the ring A⊗Z Z(0) is a central simple algebra over Z(0) , the field of fractions of Z.[1]

195.1 References [1] Artin 1999, Theorem V. 8.1.

• Artin, Michael (1999). “Noncommutative Rings”. Chapter V. • Formanek, Edward (1991). The polynomial identities and invariants of n×n matrices. Regional Conference Series in Mathematics 78. Providence, RI: American Mathematical Society. ISBN 0-8218-0730-7. Zbl 0714.16001. • Edward C. Posner, Prime rings satisfying a polynomial identity, Proc. Amer. Math. Soc. 11 (1960), pp. 180–183.

598

Chapter 196

Primary ideal In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yn is also an element of Q, for some n>0. For example, in the ring of integers Z, (pn ) is a primary ideal if p is a prime number. The notion of primary ideals is important in commutative ring theory because every ideal of a Noetherian ring has a primary decomposition, that is, can be written as an intersection of finitely many primary ideals. This result is known as the Lasker–Noether theorem. Consequently,[1] an irreducible ideal of a Noetherian ring is primary. Various methods of generalizing primary ideals to noncommutative rings exist[2] but the topic is most often studied for commutative rings. Therefore, the rings in this article are assumed to be commutative rings with identity.

196.1 Examples and properties • The definition can be rephrased in a more symmetric manner: an ideal q is primary if, whenever xy ∈ q , we √ √ have either x ∈ q or y ∈ q or x, y ∈ q . (Here q denotes the radical of q .) • An ideal Q of R is primary if and only if every zerodivisor in R/Q is nilpotent. (Compare this to the case of prime ideals, where P is prime if every zerodivisor in R/P is actually zero.) • Any prime ideal is primary, and moreover an ideal is prime if and only if it is primary and semiprime. • Every primary ideal is primal.[3] • If Q is a primary ideal, then the radical of Q is necessarily a prime ideal P, and this ideal is called the associated prime ideal of Q. In this situation, Q is said to be P-primary. • On the other hand, an ideal whose radical is prime is not necessarily primary: for example, if R = √ k[x, y, z]/(xy − z 2 ) , p = (x, z) , and q = p2 , then p is prime and q = p , but we have xy = z 2 ∈ p2 = q , x ̸∈ q , and y n ̸∈ q for all n > 0, so q is not primary. The primary decomposition of q is (x) ∩ (x2 , xz, y) ; here (x) is p -primary and (x2 , xz, y) is (x, y, z) -primary. • An ideal whose radical is maximal, however, is primary. • If P is a maximal prime ideal, then any ideal containing a power of P is P-primary. Not all P-primary ideals need be powers of P; for example the ideal (x, y2 ) is P-primary for the ideal P = (x, y) in the ring k[x, y], but is not a power of P. • In general powers of a prime ideal P need not be P-primary. (An example is given by taking R to be the ring k[x, y, z]/(xy − z2 ), with P the prime ideal (x, z). If Q = P 2 , then xy ∈ Q, but x is not in Q and y is not in the radical P of Q, so Q is not P-primary.) However every ideal Q with radical P is contained in a smallest P-primary ideal, consisting of all elements a such that ax is in Q for some x not in P. In particular there is a smallest P-primary ideal containing P n , called the nth symbolic power of P. 599

600

CHAPTER 196. PRIMARY IDEAL

• If A is a Noetherian ring and P a prime ideal, then the kernel of A → AP , the map from A to the localization of A at P, is the intersection of all P-primary ideals.[4]

196.2 Footnotes [1] To be precise, one usually uses this fact to prove the theorem. [2] See the references to Chatters-Hajarnavis, Goldman, Gorton-Heatherly, and Lesieur-Croisot. [3] For the proof of the second part see the article of Fuchs [4] Atiyah-Macdonald, Corollary 10.21

196.3 References • Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, p. 50, ISBN 978-0-201-40751-8 • Chatters, A. W.; Hajarnavis, C. R. (1971), “Non-commutative rings with primary decomposition”, Quart. J. Math. Oxford Ser. (2) 22: 73–83, doi:10.1093/qmath/22.1.73, ISSN 0033-5606, MR 0286822 • Goldman, Oscar (1969), “Rings and modules of quotients”, J. Algebra 13: 10–47, doi:10.1016/0021-8693(69)900040, ISSN 0021-8693, MR 0245608 • Gorton, Christine; Heatherly, Henry (2006), “Generalized primary rings and ideals”, Math. Pannon. 17 (1): 17–28, ISSN 0865-2090, MR 2215638 • On primal ideals, Ladislas Fuchs • Lesieur, L.; Croisot, R. (1963), Algèbre noethérienne non commutative (in French), Mémor. Sci. Math., Fasc. CLIV. Gauthier-Villars & Cie, Editeur -Imprimeur-Libraire, Paris, p. 119, MR 0155861

196.4 External links • Primary ideal at Encyclopaedia of Mathematics

Chapter 197

Prime element In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish prime elements from irreducible elements, a concept which is the same in UFDs but not the same in general.

197.1 Definition An element p of a commutative ring R is said to be prime if it is not zero or a unit and whenever p divides ab for some a and b in R, then p divides a or p divides b. Equivalently, an element p is prime if, and only if, the principal ideal (p) generated by p is a nonzero prime ideal.[1] Interest in prime elements comes from the Fundamental theorem of arithmetic, which asserts that each nonzero integer can be written in essentially only one way as 1 or −1 multiplied by a product of positive prime numbers. This led to the study of unique factorization domains, which generalize what was just illustrated in the integers. Being prime is relative to which ring an element is considered to be in; for example, 2 is a prime element in Z but it is not in Z[i], the ring of Gaussian integers, since 2 = (1 + i)(1 − i) and 2 does not divide any factor on the right.

197.2 Connection with prime ideals Main article: Prime ideal An ideal I in the ring R (with unity) is prime if the factor ring R/I is an integral domain. A nonzero principal ideal is prime if and only if it is generated by a prime element.

197.3 Irreducible elements Main article: Irreducible element Prime elements should not be confused with irreducible elements. In an integral domain, every prime is irreducible[2] but the converse is not true in general. However, in unique factorization domains,[3] or more generally in GCD domains, primes and irreducibles are the same.

197.4 Examples The following are examples of prime elements in rings: 601

602

CHAPTER 197. PRIME ELEMENT

• The integers ±2, ±3, ±5, ±7, ±11, ... in the ring of integers Z • the complex numbers (1 + i), 19, and (2 + 3i) in the ring of Gaussian integers Z[i] • the polynomials x2 − 2 and x2 + 1 in Z[x], the ring of polynomials over Z.

197.5 References Notes [1] Hungerford 1980, Theorem III.3.4(i), as indicated in the remark below the theorem and the proof, the result holds in full generality. [2] Hungerford 1980, Theorem III.3.4(iii) [3] Hungerford 1980, Remark after Definition III.3.5

Sources • Section III.3 of Hungerford, Thomas W. (1980), Algebra, Graduate Texts in Mathematics 73 (Reprint of 1974 ed.), New York: Springer-Verlag, ISBN 978-0-387-90518-1, MR 0600654 • Jacobson, Nathan (1989), Basic algebra. II (2 ed.), New York: W. H. Freeman and Company, pp. xviii+686, ISBN 0-7167-1933-9, MR 1009787 • Kaplansky, Irving (1970), Commutative rings, Boston, Mass.: Allyn and Bacon Inc., pp. x+180, MR 0254021

Chapter 198

Prime ideal This article is about ideals in ring theory. For prime ideals in order theory, see ideal (order theory)#Prime ideals. In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring

A Hasse diagram of a portion of the lattice of ideals of the integers Z. The purple nodes indicate prime ideals. The purple and green nodes are semiprime ideals, and the purple and blue nodes are primary ideals.

of integers.[1][2] The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal. Primitive ideals are prime, and prime ideals are both primary and semiprime.

198.1 Prime ideals for commutative rings An ideal P of a commutative ring R is prime if it has the following two properties: • If a and b are two elements of R such that their product ab is an element of P, then a is in P or b is in P, • P is not equal to the whole ring R. This generalizes the following property of prime numbers: if p is a prime number and if p divides a product ab of two integers, then p divides a or p divides b. We can therefore say A positive integer n is a prime number if and only if the ideal nZ is a prime ideal in Z. 603

604

198.1.1

CHAPTER 198. PRIME IDEAL

Examples

• If R denotes the ring C[X, Y] of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial Y 2 − X 3 − X − 1 is a prime ideal (see elliptic curve). • In the ring Z[X] of all polynomials with integer coefficients, the ideal generated by 2 and X is a prime ideal. It consists of all those polynomials whose constant coefficient is even. • In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i.e. M is contained in exactly two ideals of R, namely M itself and the entire ring R. Every maximal ideal is in fact prime. In a principal ideal domain every nonzero prime ideal is maximal, but this is not true in general. • If M is a smooth manifold, R is the ring of smooth real functions on M, and x is a point in M, then the set of all smooth functions f with f (x) = 0 forms a prime ideal (even a maximal ideal) in R.

198.1.2

Properties

• An ideal I in the ring R (with unity) is prime if and only if the factor ring R/I is an integral domain. In particular, a commutative ring is an integral domain if and only if {0} is a prime ideal. • An ideal I is prime if and only if its set-theoretic complement is multiplicatively closed.[3] • Every nonzero ring contains at least one prime ideal (in fact it contains at least one maximal ideal), which is a direct consequence of Krull’s theorem. • The set of all prime ideals (the spectrum of a ring) contains minimal elements (called minimal prime). Geometrically, these correspond to irreducible components of the spectrum. • The preimage of a prime ideal under a ring homomorphism is a prime ideal. • The sum of two prime ideals is not necessarily prime. For an example, consider the ring C[x, y] with prime ideals P = (x2 + y2 − 1) and Q = (x) (the ideals generated by x2 + y2 − 1 and x respectively). Their sum P + Q = (x2 + y2 − 1, x) = (y2 − 1, x) however is not prime: y2 − 1 = (y − 1)(y + 1) ∈ P + Q but its two factors are not. Alternatively, note that the quotient ring has zero divisors so it is not an integral domain and thus P + Q cannot be prime. • In a commutative ring R with at least two elements, if every proper ideal is prime, then the ring is a field. (If the ideal (0) is prime, then the ring R is an integral domain. If q is any non-zero element of R and the ideal (q2 ) is prime, then it contains q and then q is invertible.) • A nonzero principal ideal is prime if and only if it is generated by a prime element. In a UFD, every nonzero prime ideal contains a prime element.

198.1.3

Uses

One use of prime ideals occurs in algebraic geometry, where varieties are defined as the zero sets of ideals in polynomial rings. It turns out that the irreducible varieties correspond to prime ideals. In the modern abstract approach, one starts with an arbitrary commutative ring and turns the set of its prime ideals, also called its spectrum, into a topological space and can thus define generalizations of varieties called schemes, which find applications not only in geometry, but also in number theory. The introduction of prime ideals in algebraic number theory was a major step forward: it was realized that the important property of unique factorisation expressed in the fundamental theorem of arithmetic does not hold in every ring of algebraic integers, but a substitute was found when Richard Dedekind replaced elements by ideals and prime elements by prime ideals; see Dedekind domain.

198.2. PRIME IDEALS FOR NONCOMMUTATIVE RINGS

605

198.2 Prime ideals for noncommutative rings The notion of a prime ideal can be generalized to noncommutative rings by using the commutative definition “idealwise”. Wolfgang Krull advanced this idea in 1928.[4] The following content can be found in texts such as (Goodearl 2004) and (Lam 2001). If R is a (possibly noncommutative) ring and P is an ideal in R other than R itself, we say that P is prime if for any two ideals A and B of R: • If the product of ideals AB is contained in P, then at least one of A and B is contained in P. It can be shown that this definition is equivalent to the commutative one in commutative rings. It is readily verified that if an ideal of a noncommutative ring R satisfies the commutative definition of prime, then it also satisfies the noncommutative version. An ideal P satisfying the commutative definition of prime is sometimes called a completely prime ideal to distinguish it from other merely prime ideals in the ring. Completely prime ideals are prime ideals, but the converse is not true. For example, the zero ideal in the ring of n × n matrices over a field is a prime ideal, but it is not completely prime. This is close to the historical point of view of ideals as ideal numbers, as for the ring Z “A is contained in P” is another way of saying “P divides A”, and the unit ideal R represents unity. Equivalent formulations of the ideal P ≠ R being prime include the following properties: • For all a and b in R, (a)(b) ⊆ P implies a ∈ P or b ∈ P. • For any two right ideals of R, AB ⊆ P implies A ⊆ P or B ⊆ P. • For any two left ideals of R, AB ⊆ P implies A ⊆ P or B ⊆ P. • For any elements a and b of R, if aRb ⊆ P, then a ∈ P or b ∈ P. Prime ideals in commutative rings are characterized by having multiplicatively closed complements in R, and with slight modification, a similar characterization can be formulated for prime ideals in noncommutative rings. A nonempty subset S ⊆ R is called an m-system if for any a and b in S, there exists r in R such that arb is in S.[5] The following item can then be added to the list of equivalent conditions above: • The complement R\P is an m-system.

198.2.1

Examples

• Any primitive ideal is prime. • As with commutative rings, maximal ideals are prime, and also prime ideals contain minimal prime ideals. • A ring is a prime ring if and only if the zero ideal is a prime ideal, and moreover a ring is a domain if and only if the zero ideal is a completely prime ideal. • Another fact from commutative theory echoed in noncommutative theory is that if A is a nonzero R module, and P is a maximal element in the poset of annihilator ideals of submodules of A, then P is prime.

198.3 Important facts • Prime avoidance lemma. If R is a commutative ring, and A is a subring (possibly without unity), and I 1 , ..., In is a collection of ideals of R with at most two members not prime, then if A is not contained in any Ij, it is also not contained in the union of I 1 , ..., In.[6] In particular, A could be an ideal of R. • If S is any m-system in R, then a lemma essentially due to Krull shows that there exists an ideal of R maximal with respect to being disjoint from S, and moreover the ideal must be prime.[7] In the case {S} = {1}, we have Krull’s theorem, and this recovers the maximal ideals of R. Another prototypical m-system is the set, {x, x2 , x3 , x4 , ...}, of all positive powers of a non-nilpotent element.

606

CHAPTER 198. PRIME IDEAL

• For a prime ideal P, the complement R\P has another property beyond being an m-system. If xy is in R\P, then both x and y must be in R\P, since P is an ideal. A set that contains the divisors of its elements is called saturated. • For a commutative ring R, there is a kind of converse for the previous statement: If S is any nonempty saturated and multiplicatively closed subset of R, the complement R\S is a union of prime ideals of R.[8] • The intersection of members of a descending chain of prime ideals is a prime ideal, and in a commutative ring the union of members of an ascending chain of prime ideals is a prime ideal. With Zorn’s Lemma, these observations imply that the poset of prime ideals of a commutative ring (partially ordered by inclusion) has maximal and minimal elements.

198.4 Connection to maximality Prime ideals can frequently be produced as maximal elements of certain collections of ideals. For example: • An ideal maximal with respect to having empty intersection with a fixed m-system is prime. • An ideal maximal among annihilators of submodules of a fixed R module M is prime. • In a commutative ring, an ideal maximal with respect to being non-principal is prime.[9] • In a commutative ring, an ideal maximal with respect to being not countably generated is prime.[10]

198.5 References [1] Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9. [2] Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X. [3] Reid, Miles (1996). Undergraduate Commutative Algebra. Cambridge University Press. ISBN 0-521-45889-7. [4] Krull, Wolfgang, Primidealketten in allgemeinen Ringbereichen, Sitzungsberichte Heidelberg. Akad. Wissenschaft (1928), 7. Abhandl.,3-14. [5] Obviously, multiplicatively closed sets are m-systems. [6] Jacobson Basic Algebra II, p. 390 [7] Lam First Course in Noncommutative Rings, p. 156 [8] Kaplansky Commutative rings, p. 2 [9] Kaplansky Commutative rings, p. 10, Ex 10. [10] Kaplansky Commutative rings, p. 10, Ex 11.

198.6 Further reading • Goodearl, K. R.; Warfield, R. B., Jr. (2004), An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts 61 (2 ed.), Cambridge: Cambridge University Press, pp. xxiv+344, ISBN 0-521-54537-4, MR 2080008 • Jacobson, Nathan (1989), Basic algebra. II (2 ed.), New York: W. H. Freeman and Company, pp. xviii+686, ISBN 0-7167-1933-9, MR 1009787 • Kaplansky, Irving (1970), Commutative rings, Boston, Mass.: Allyn and Bacon Inc., pp. x+180, MR 0254021

198.6. FURTHER READING

607

• Lam, T. Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics 131 (2nd ed.), New York: Springer-Verlag, pp. xx+385, ISBN 0-387-95183-0, MR 1838439, Zbl 0980.16001 • Lam, T. Y.; Reyes, Manuel L. (2008), “A prime ideal principle in commutative algebra”, J. Algebra 319 (7): 3006–3027, doi:10.1016/j.jalgebra.2007.07.016, ISSN 0021-8693, MR 2397420, Zbl 1168.13002 • Hazewinkel, Michiel, ed. (2001), “Prime ideal”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608010-4

Chapter 199

Prime ring In abstract algebra, a nonzero ring R is a prime ring if for any two elements a and b of R, arb = 0 for all r in R implies that either a = 0 or b = 0. This definition can be regarded as a simultaneous generalization of both integral domains and simple rings. Prime ring can also refer to the subring of a field determined by its characteristic. For a characteristic 0 field, the prime ring is the integers, for a characteristic p field (with p a prime number) the prime ring is the finite field of order p (cf. prime field).[1]

199.1 Equivalent definitions A ring R is prime if and only if the zero ideal {0} is a prime ideal in the noncommutative sense. This being the case, the equivalent conditions for prime ideals yield the following equivalent conditions for R to be a prime ring: • For any two ideals A and B of R, AB={0} implies A={0} or B={0}. • For any two right ideals A and B of R, AB={0} implies A={0} or B={0}. • For any two left ideals A and B of R, AB={0} implies A={0} or B={0}. Using these conditions it can be checked that the following are equivalent to R being a prime ring: • All right ideals are faithful modules as right R modules. • All left ideals are faithful left R modules.

199.2 Examples • Any domain is a prime ring. • Any simple ring is a prime ring, and more generally: every left or right primitive ring is a prime ring. • Any matrix ring over an integral domain is a prime ring. In particular, the ring of 2-by-2 integer matrices is a prime ring.

199.3 Properties • A commutative ring is a prime ring if and only if it is an integral domain. • A ring is prime if and only if its zero ideal is a prime ideal. 608

199.4. NOTES

609

• A nonzero ring is prime if and only if the monoid of its ideals lacks zero divisors. • The ring of matrices over a prime ring is again a prime ring.

199.4 Notes [1] Page 90 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-201-555400, Zbl 0848.13001

199.5 References • Lam, Tsit-Yuen (2001), A First Course in Noncommutative Rings (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-95325-0, MR 1838439

Chapter 200

Primitive ideal Not to be confused with primary ideal. In mathematics, a left primitive ideal in ring theory is the annihilator of a simple left module. A right primitive ideal is defined similarly. Note that (despite the name) left and right primitive ideals are always two-sided ideals. The quotient of a ring by a left primitive ideal is a left primitive ring.

200.1 References • Isaacs, I. Martin (1994), Algebra, Brooks/Cole Publishing Company, ISBN 0-534-19002-2

610

Chapter 201

Primitive ring In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero.

201.1 Definition A ring R is said to be a left primitive ring if and only if it has a faithful simple left R-module. A right primitive ring is defined similarly with right R-modules. There are rings which are primitive on one side but not on the other. The first example was constructed by George M. Bergman in (Bergman 1964). Another example found by Jategaonkar showing the distinction can be found in (Rowen 1988, p.159) An internal characterization of left primitive rings is as follows: a ring is left primitive if and only if there is a maximal left ideal containing no nonzero two-sided ideals. The analogous definition for right primitive rings is also valid. The structure of left primitive rings is completely determined by the Jacobson density theorem: A ring is left primitive if and only if it is isomorphic to a dense subring of the ring of endomorphisms of a left vector space over a division ring. Another equivalent definition states that a ring is left primitive if and only if it is a prime ring with a faithful left module of finite length (Lab 2001, Ex. 11.19, p. 191).

201.2 Properties One sided primitive rings are both semiprimitive rings and prime rings. Since the ring product of two or more nonzero rings is not prime, it is clear that the product of primitive rings is never primitive. For a left Artinian ring, it is known that the conditions “left primitive”, “right primitive”, “prime”, and "simple" are all equivalent, and in this case it is a semisimple ring isomorphic to a square matrix ring over a division ring. More generally, in any ring with a minimal one sided ideal, “left primitive"="right primitive"="prime”. A commutative ring is left primitive if and only if it is a field. Being left primitive is a Morita invariant property.

201.3 Examples Every simple ring R with unity is both left and right primitive. (However, a simple non-unital ring may not be primitive.) This follows from the fact that R has a maximal left ideal M, and the fact that the quotient module R/M is a simple left R-module, and that its annihilator is a proper two-sided ideal in R. Since R is a simple ring, this annihilator is {0} and therefore R/M is a faithful left R-module. 611

612

CHAPTER 201. PRIMITIVE RING

Weyl algebras over fields with characteristic zero are primitive, and since they are domains, they are examples without minimal one-sided ideals.

201.3.1

Full linear rings

A special case of primitive rings is that of full linear rings. A left full linear ring is the ring of all linear transformations of an infinite dimensional left vector space over a division ring. (A right full linear ring differs by using a right vector space instead.) In symbols, R = End(D V ) where V is a vector space over a division ring D. It is known that R is a left full linear ring if and only if R is von Neumann regular, left self-injective with socle soc(RR)≠{0}. (Goodearl 1991, p. 100) Through linear algebra arguments, it can be shown that End(D V ) is isomorphic to the ring of row finite matrices RFMI (D) , where I is an index set whose size is the dimension of V over D. Likewise right full linear rings can be realized as column finite matrices over D. Using this we can see that there are non-simple left primitive rings. By the Jacobson Density characterization, a left full linear ring R is always left primitive. When dimDV is finite R is a square matrix ring over D, but when dimDV is infinite, the set of finite rank linear transformations is a proper two-sided ideal of R, and hence R is not simple.

201.4 References • Bergman, G. M. (1964), “A ring primitive on the right but not on the left”, Proceedings of the American Mathematical Society (American Mathematical Society) 15 (3): 473–475, doi:10.1090/S0002-9939-1964-01674974, ISSN 0002-9939, JSTOR 2034527, MR 0167497 p. 1000 errata • Goodearl, K. R. (1991), von Neumann regular rings (2 ed.), Malabar, FL: Robert E. Krieger Publishing Co. Inc., pp. xviii+412, ISBN 0-89464-632-X, MR 1150975 (93m:16006)} • Lam, Tsi-Yuen (2001), A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131 (2nd ed.), Springer, ISBN 9781441986160, MR 1838439 • Rowen, Louis H. (1988), Ring theory. Vol. I, Pure and Applied Mathematics 127, Boston, MA: Academic Press Inc., pp. xxiv+538, ISBN 0-12-599841-4, MR 940245 (89h:16001)

Chapter 202

Principal ideal A principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R.

202.1 Definitions • a left principal ideal of R is a subset of R of the form Ra = {ra : r in R}; • a right principal ideal is a subset of the form aR = {ar : r in R}; • a two-sided principal ideal is a subset of the form RaR = {r1 as1 + ... + rnasn : r1 ,s1 ,...,rn,sn in R}. If R is a commutative ring, then the above three notions are all the same. In that case, it is common to write the ideal generated by a as ⟨a⟩.

202.2 Examples of non-principal ideal Not all ideals are principal. For example, consider the commutative ring C[x,y] of all polynomials in two variables x and y, with complex coefficients. The ideal ⟨x,y⟩ generated by x and y, which consists of all the polynomials in C[x,y] that have zero for the constant term, is not principal. To see this, suppose that p were a generator for ⟨x,y⟩; then x and y would both be divisible by p, which is impossible unless p is a nonzero constant. But zero is the only constant in ⟨x,y⟩, so we have a contradiction.

202.3 Related definitions A ring in which every ideal is principal is called principal, or a principal ideal ring. A principal ideal domain (PID) is an integral domain that is principal. Any PID must be a unique factorization domain; the normal proof of unique factorization in the integers (the so-called fundamental theorem of arithmetic) holds in any PID.

202.4 Properties Any Euclidean domain is a PID; the algorithm used to calculate greatest common divisors may be used to find a generator of any ideal. More generally, any two principal ideals in a commutative ring have a greatest common divisor in the sense of ideal multiplication. In principal ideal domains, this allows us to calculate greatest common divisors of elements of the ring, up to multiplication by a unit; we define gcd(a,b) to be any generator of the ideal ⟨a,b⟩. For a Dedekind domain R, we may also ask, given a non-principal ideal I of R, whether there is some extension S of R such that the ideal of S generated by I is principal (said more loosely, I becomes principal in S). This question arose 613

614

CHAPTER 202. PRINCIPAL IDEAL

in connection with the study of rings of algebraic integers (which are examples of Dedekind domains) in number theory, and led to the development of class field theory by Teiji Takagi, Emil Artin, David Hilbert, and many others. The principal ideal theorem of class field theory states that every integer ring R (i.e. the ring of integers of some number field) is contained in a larger integer ring S which has the property that every ideal of R becomes a principal ideal of S. In this theorem we may take S to be the ring of integers of the Hilbert class field of R; that is, the maximal unramified abelian extension (that is, Galois extension whose Galois group is abelian) of the fraction field of R, and this is uniquely determined by R. Krull’s principal ideal theorem states that if R is a Noetherian ring and I is a principal, proper ideal of R, then I has height at most one.

202.5 See also • Ascending chain condition for principal ideals

202.6 References • Joseph A. Gallian (2004). Contemporary Abstract Algebra. Houghton Mifflin. p. 262. ISBN 978-0-61851471-7.

Chapter 203

Principal ideal domain In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors (e.g., Bourbaki) refer to PIDs as principal rings. The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot. Principal ideal domains are thus mathematical objects that behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements (so an analogue of the fundamental theorem of arithmetic holds); any two elements of a PID have a greatest common divisor (although it may not be possible to find it using the Euclidean algorithm). If x and y are elements of a PID without common divisors, then every element of the PID can be written in the form ax + by. Principal ideal domains are noetherian, they are integrally closed, they are unique factorization domains and Dedekind domains. All Euclidean domains and all fields are principal ideal domains. Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields

203.1 Examples Examples include: • K: any field, • Z: the ring of integers,[1] • K[x]: rings of polynomials in one variable with coefficients in a field. (The converse is also true; that is, if A[x] is a PID, then A is a field.) Furthermore, a ring of formal power series in one variable over a field is a PID since every ideal is of the form (xk ) . • Z[i]: the ring of Gaussian integers[2] • Z[ω] (where ω is a primitive cube root of 1): the Eisenstein integers Examples of integral domains that are not PIDs: • Z[x]: the ring of all polynomials with integer coefficients --- it is not principal because the ideal generated by 2 and X is an example of an ideal that cannot be generated by a single polynomial. • K[x,y]: The ideal (x,y) is not principal. 615

616

CHAPTER 203. PRINCIPAL IDEAL DOMAIN

203.2 Modules Main article: Structure theorem for finitely generated modules over a principal ideal domain The key result is the structure theorem: If R is a principal ideal domain, and M is a finitely generated R-module, then M is a direct sum of cyclic modules, i.e., modules with one generator. The cyclic modules are isomorphic to R/xR for some x ∈ R [3] (notice that x may be equal to 0 , in which case R/xR is R ). If M is a free module over a principal ideal domain R, then every submodule of M is again free. This does not hold for modules over arbitrary rings, as the example (2, X) ⊆ Z[X] of modules over Z[X] shows.

203.3 Properties In a principal ideal domain, any two elements a,b have a greatest common divisor, which may be obtained as a generator of the ideal (a,b). All Euclidean domains are principal ideal domains, but [ the√converse ]is not true. An example of a principal ideal domain that is not a Euclidean domain is the ring Z (1 + −19)/2 . [4][5] In this domain no q and r exist, with √ √ 0≤|r|1.

• Commutative Artinian serial rings are all Frobenius, and in fact have the additional property that every quotient ring R/I is also Frobenius. It turns out that among commutative Artinian rings, the serial rings are exactly the rings whose (nonzero) quotients are all Frobenius. • Many exotic PF and FPF rings can be found as examples in (Faith 1984)

216.4 See also • Quasi-Frobenius Lie algebra

216.5. NOTES

653

216.5 Notes The definitions for QF, PF and FPF are easily seen to be categorical properties, and so they are preserved by Morita equivalence, however being a Frobenius ring is not preserved. For one-sided Noetherian rings the conditions of left or right PF both coincide with QF, but FPF rings are still distinct. A finite-dimensional algebra R over a field k is a Frobenius k-algebra if and only if R is a Frobenius ring. QF rings have the property that all of their modules can be embedded in a free R module. This can be seen in the following way. A module M embeds into its injective hull E(M), which is now also projective. As a projective module, E(M) is a summand of a free module F, and so E(M) embeds in F with the inclusion map. By composing these two maps, M is embedded in F.

216.6 Textbooks • Anderson, Frank Wylie; Fuller, Kent R (1992), Rings and Categories of Modules, Berlin, New York: SpringerVerlag, ISBN 978-0-387-97845-1 • Faith, Carl; Page, Stanley (1984), FPF Ring Theory: Faithful modules and generators of Mod-$R$, London Mathematical Society Lecture Note Series No. 88, Cambridge University Press, ISBN 0-521-27738-8, MR 0754181 • Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294 • Nicholson, W. K.; Yousif, M. F. (2003), Quasi-Frobenius rings, Cambridge University Press, ISBN 0-52181593-2

216.7 References For QF-1, QF-2, QF-3 rings: • Morita, Kiiti (1958), “On algebras for which every faithful representation is its own second commutator”, Math. Z. 69: 429–434, doi:10.1007/bf01187420, ISSN 0025-5874 • Ringel, Claus Michael; Tachikawa, Hiroyuki (1974), "${\rm QF}−3$ rings”, J. Reine Angew. Math. 272: 49–72, ISSN 0075-4102 • Thrall, R. M. (1948), “Some generalization of quasi-Frobenius algebras”, Trans. Amer. Math. Soc. 64: 173– 183, doi:10.1090/s0002-9947-1948-0026048-0, ISSN 0002-9947

Chapter 217

Quasiregular element This article addresses the notion of quasiregularity in the context of ring theory, a branch of modern algebra. For other notions of quasiregularity in mathematics, see the disambiguation page quasiregular. In mathematics, specifically ring theory, the notion of quasiregularity provides a computationally convenient way to work with the Jacobson radical of a ring.[1] Intuitively, quasiregularity captures what it means for an element of a ring to be “bad"; that is, have undesirable properties.[2] Although a “bad element” is necessarily quasiregular, quasiregular elements need not be “bad,” in a rather vague sense. In this article, we primarily concern ourselves with the notion of quasiregularity for unital rings. However, one section is devoted to the theory of quasiregularity in non-unital rings, which constitutes an important aspect of noncommutative ring theory.

217.1 Definition Let R be a ring (with unity) and let r be an element of R. Then r is said to be quasiregular, if 1 − r is a unit in R; that is, invertible under multiplication.[1] The notions of right or left quasiregularity correspond to the situations where 1 − r has a right or left inverse, respectively.[1] An element x of a non-unital ring is said to be right quasiregular if there is y such that x+y −xy = 0 .[3] The notion of a left quasiregular element is defined in an analogous manner. The element y is sometimes referred to as a right quasi-inverse of x.[4] If the ring is unital, this definition quasiregularity coincides with that given above.[5] If one writes x · y = x + y − xy , then this binary operation · is associative.[6] In fact, the map (R, ·) → (R, ×); x 7→ 1 − x (where × denotes the multiplication of the ring R) is a monoid isomorphism.[5] Therefore, if an element possesses both a left and right quasi-inverse, they are equal.[7] Note that some authors use different definitions. They call an element x right quasiregular if there exists y such that x + y + xy = 0 ,[8] which is equivalent to saying that 1 + x has a right inverse when the ring is unital. If we write x ◦ y = x + y + xy , then (−x) ◦ (−y) = −(x · y) , so we can easily go from one set-up to the other by changing signs.[9] For example, x is right quasiregular in one set-up iff −x is right quasiregular in the other set-up.[9]

217.2 Examples • If R is a rng, then the additive identity of R is always quasiregular. • If x2 is right (resp. left) quasiregular, then x is right (resp. left) quasiregular.[10] • If R is a rng, every nilpotent element of R is quasiregular.[11] This fact is supported by an elementary computation: xn+1 = 0 (1 − x)(1 + x + x2 + · · · + xn ) = 1 (or (1 + x)(1 − x + x2 − · · · + (−x)n ) = 1 if we follow the second convention). From this we see easily that the quasi-inverse of x is −x − x2 − · · · − xn (or −x + x2 − · · · + (−x)n ). 654

217.3. PROPERTIES

655

• In the second convention, a matrix is quasiregular in a matrix ring if it does not possess −1 as an eigenvalue. More generally, a bounded operator is quasiregular if −1 is not in its spectrum. • In a unital Banach algebra, if ∥x∥ < 1 , then the geometric series such x is quasiregular.

∑∞ 0

xn converges. Consequently, every

• If R is a ring and S=R[[''X''1,...,''X''n]] denotes the ring of formal power series in n intederminants over R, an element of S is quasiregular if and only its constant term is quasiregular as an element of R.

217.3 Properties • Every element of the Jacobson radical of a (not necessarily commutative) ring is quasiregular.[12] In fact, the Jacobson radical of a ring can be characterized as the unique right ideal of the ring, maximal with respect to the property that every element is right quasiregular.[13][14] However, a right quasiregular element need not necessarily be a member of the Jacobson radical.[15] This justifies the remark in the beginning of the article - “bad elements” are quasiregular, although quasiregular elements are not necessarily “bad.” Elements of the Jacobson radical of a ring, are often deemed to be “bad.” • If an element of a ring is nilpotent and central, then it is a member of the ring’s Jacobson radical.[16] This is because the principal right ideal generated by that element consists of quasiregular (in fact, nilpotent) elements only. • If an element, r, of a ring is idempotent, it cannot be a member of the ring’s Jacobson radical.[17] This is because idempotent elements cannot be quasiregular. This property, as well as the one above, justify the remark given at the top of the article that the notion of quasiregularity is computationally convenient when working with the Jacobson radical.[1]

217.4 Generalization to semirings The notion of quasiregular element readily generalizes to semirings. If a is an element of a semiring S, then an affine map from S to itself is µa (r) = ra + 1 . An element a of S is said to be right quasiregular if µa has a fixed point, which need not be unique. Each such fixed point is called a left quasi-inverse of a. If b is a left quasi-inverse of a and additionally b = ab + 1, then b it is called a quasi-inverse of a; any element of the semiring that has a quasi-inverse is said to be quasiregular. It is possible that some but not all elements of a semiring be quasiregular; for example, 1 in the semiring of nonegative reals with the usual addition and multiplication of reals, µa has the fixed point 1−a for all a < 1, but has no fixed point for a ≥ 1.[18] If every element of a semiring is quasiregular then the semiring is called a quasi-regular semiring, closed semiring,[19] or occasionally a Lehmann semiring[18] (the latter honoring the paper of Daniel J. Lehmann.[20] ) Examples of quasi-regular semirings are provided by the Kleene algebras (prominently among them, the algebra of regular expressions), in which the quasi-inverse is lifted to the role of a unary operation (denoted by a*) defined as the least fixedpoint solution. Kleene algebras are additively idempotent but not all quasi-regular semirings are so. We can extend the example of nonegative reals to include infinity and it becomes a quasi-regular semiring with the quasiinverse of any element a ≥ 1 being the infinity. This quasi-regular semiring is not additively idempotent however, so it is not a Kleene algebra.[19] It is however a complete semiring.[21] More generally, all complete semirings are quasiregular.[22] The term closed semiring is actually used by some authors to mean complete semiring rather than just quasiregular.[23][24] Conway semirings are also quasiregular; the two Conway axioms are actually independent, i.e. there are semirings satisfying only the product-star [Conway] axiom, (ab)* = 1+a(ba)*b, but not the sum-star axiom, (a+b)* = (a*b)*a* and vice versa; it is the product-star [Conway] axiom that implies that a semiring is quasiregular. Additionally, a commutative semiring is quasiregular if and only if it satisfies the product-star Conway axiom.[18] Quasiregular semirings appear in algebraic path problems, a generalization of the shortest path problem.[19]

656

CHAPTER 217. QUASIREGULAR ELEMENT

217.5 See also • inverse element

217.6 Notes [1] Isaacs, p. 180 [2] Isaacs, p. 179 [3] Lam, Ex. 4.2, p. 50 [4] Polcino & Sehgal (2002), p. 298. [5] Lam, Ex. 4.2(3), p. 50 [6] Lam, Ex. 4.1, p. 50 [7] Since 0 is the multiplicative identity, if x · y = 0 = y ′ · x , then y = (y ′ · x) · y = y ′ · (x · y) = y ′ . Quasiregularity does not require the ring to have a multiplicative identity. [8] Kaplansky, p. 85 [9] Lam, p. 51 [10] Kaplansky, p. 108 [11] Lam, Ex. 4.2(2), p. 50 [12] Isaacs, Theorem 13.4(a), p. 180 [13] Isaacs, Theorem 13.4(b), p. 180 [14] Isaacs, Corollary 13.7, p. 181 [15] Isaacs, p. 181 [16] Isaacs, Corollary 13.5, p. 181 [17] Isaacs, Corollary 13.6, p. 181 [18] Jonathan S. Golan (30 June 2003). Semirings and Affine Equations over Them. Springer Science & Business Media. pp. 157–159 and 164–165. ISBN 978-1-4020-1358-4. [19] Marc Pouly; Jürg Kohlas (2011). Generic Inference: A Unifying Theory for Automated Reasoning. John Wiley & Sons. pp. 232 and 248–249. ISBN 978-1-118-01086-0. [20] Lehmann, D. J. (1977). “Algebraic structures for transitive closure”. Theoretical Computer Science 4: 59. doi:10.1016/03043975(77)90056-1. [21] Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. Handbook of Weighted Automata, 3–28. doi:10.1007/9783-642-01492-5_1, pp. 7-10 [22] U. Zimmermann (1981). Linear and combinatorial optimization in ordered algebraic structures. Elsevier. p. 141. ISBN 978-0-08-086773-1. [23] Dexter Kozen (1992). The Design and Analysis of Algorithms. Springer Science & Business Media. p. 31. ISBN 978-0387-97687-7. [24] J.A. Storer (2001). An Introduction to Data Structures and Algorithms. Springer Science & Business Media. p. 336. ISBN 978-0-8176-4253-2.

217.7. REFERENCES

657

217.7 References • I. Martin Isaacs (1993). Algebra, a graduate course (1st ed.). Brooks/Cole Publishing Company. ISBN 0-53419002-2. • Irving Kaplansky (1969). Fields and Rings. The University of Chicago Press. • Lam, Tsit-Yuen (2003). Exercises in Classical Ring Theory. Problem Books in Mathematics (2nd ed.). SpringerVerlag. ISBN 978-0387005003. • Milies, César Polcino; Sehgal, Sudarshan K. (2002). An introduction to group rings. Springer. ISBN 978-14020-0238-0.

217.8 See also • Jacobson radical • Nilradical • Unit (ring theory) • Nilpotent element • Center of a ring • Idempotent element

Chapter 218

Quasisymmetric function For quasisymmetric functions in the theory of metric spaces or complex analysis, see quasisymmetric map. In algebra and in particular in algebraic combinatorics, a quasisymmetric function is any element in the ring of quasisymmetric functions which is in turn a subring of the formal power series ring with a countable number of variables. This ring generalizes the ring of symmetric functions. This ring can be realized as a specific limit of the rings of quasisymmetric polynomials in n variables, as n goes to infinity. This ring serves as universal structure in which relations between quasisymmetric polynomials can be expressed in a way independent of the number n of variables (but its elements are neither polynomials nor functions).

218.1 Definitions The ring of quasisymmetric functions, denoted QSym, can be defined over any commutative ring R such as the integers. Quasisymmetric functions are power series of bounded degree in variables x1 , x2 , x3 , . . . with coefficients in αk 1 α2 R, which are shift invariant in the sense that the coefficient of the monomial xα 1 x2 · · · xk is equal to the coefficient αk α1 α2 of the monomial xi1 xi2 · · · xik for any strictly increasing sequence of positive integers i1 < i2 < · · · < ik indexing the variables and any positive integer sequence (α1 , α2 , . . . , αk ) of exponents.[1] Much of the study of quasisymmetric functions is based on that of symmetric functions. A quasisymmetric function in finitely many variables is a quasisymmetric polynomial. Both symmetric and quasisymmetric polynomials may be characterized in terms of actions of the symmetric group Sn on a polynomial ring in n variables x1 , . . . , xn . One such action of Sn permutes variables, changing a polynomial p(x1 , . . . , xn ) by iteratively swapping pairs (xi , xi+1 ) of variables having consecutive indices. Those polynomials unchanged by all such swaps form the subring of symmetric polynomials. A second action of Sn conditionally permutes variables, changing a polynomial p(x1 , . . . , xn ) by swapping pairs (xi , xi+1 ) of variables except in monomials containing both variables. Those polynomials unchanged by all such conditional swaps form the subring of quasisymmetric polynomials. One quasisymmetric function in four variables is the polynomial

x21 x2 x3 + x21 x2 x4 + x21 x3 x4 + x22 x3 x4 . The simplest symmetric function containing all of these monomials is

x21 x2 x3 + x21 x2 x4 + x21 x3 x4 + x22 x3 x4 + x1 x22 x3 + x1 x22 x4 + x1 x23 x4 + x2 x23 x4 + x1 x2 x23 + x1 x2 x24 + x1 x3 x24 + x2 x3 x24 .

218.2 Important bases QSym is a graded R-algebra, decomposing as 658

218.3. APPLICATIONS

QSym =

⊕

659

QSymn ,

n≥0

where QSymn is the R -span of all quasisymmetric functions that are homogeneous of degree n . Two natural bases for QSymn are the monomial basis {Mα } and the fundamental basis {Fα } indexed by compositions α = (α1 , α2 , . . . , αk ) of n , denoted α ⊨ n . The monomial basis consists of M0 = 1 and all formal power series ∑

Mα =

αk 2 xiα11 xα i2 · · · xik .

i1 θ/2 741

742

a′′

CHAPTER 241. ROTOR (MATHEMATICS)

/2= +

m

a

n

a′

α < θ/2 Rotation of a vector a through angle θ, as a double reflection along two unit vectors n and m, separated by angle θ/2 (not just θ). Each prime on a indicates a reflection. The plane of the diagram is the plane of rotation. Reflections along a vector in geometric algebra may be represented as (minus) sandwiching a multivector M between a non-null vector v perpendicular to the hyperplane of reflection and that vector’s inverse v−1 : −vM v −1 and are of even grade. Under a rotation generated by the rotor R, a general multivector M will transform doublesidedly as RM R−1 .

241.1.2

Restricted alternative formulation

For a Euclidean space, it may be convenient to consider an alternative formulation, and some authors define the operation of reflection as (minus) the sandwiching of a unit (i.e. normalized) multivector: −vM v,

v 2 = 1,

forming rotors that are automatically normalised: ˜ = RR ˜ = 1. RR The derived rotor action is then expressed as a sandwich product with the reverse: ˜ RM R For a reflection for which the associated vector squares to a negative scalar, as may be the case with a pseudo-Euclidean space, such a vector can only be normalized up to the sign of its square, and additional bookkeeping of the sign of the application the rotor becomes necessary. The formulation in terms of the sandwich product with the inverse as above suffers no such shortcoming.

241.1.3

Rotations of multivectors and spinors

However, though as multivectors rotors also transform double-sidedly, rotors can be combined and form a group, and so multiple rotors compose single-sidedly. The alternative formulation above is not self-normalizing and motivates

241.2. HOMOGENEOUS REPRESENTATION ALGEBRAS

743

the definition of spinor in geometric algebra as an object that transforms single-sidedly – i.e. spinors may be regarded as non-normalised rotors in which the reverse rather than the inverse is used in the sandwich product.

241.2 Homogeneous representation algebras In homogeneous representation algebras such as conformal geometric algebra, a rotor in the representation space corresponds to a rotation about an arbitrary point, a translation or possibly another transformation in the base space.

241.3 See also • Double rotation • Lie group • Euler’s formula • Generator (mathematics)

Chapter 242

SBI ring In algebra, an SBI ring is a ring R (with identity) such that every idempotent of R modulo the Jacobson radical can be lifted to R. The abbreviation SBI was introduced by Irving Kaplansky and stands for “suitable for building idempotent elements” (Jacobson 1956, p.53).

242.1 Examples • Any ring with nil radical is SBI. • Any Banach algebra is SBI: more generally, so is any compact topological ring. • The ring of rational numbers with odd denominator is not SBI.

242.2 References • Jacobson, Nathan (1956), Structure of rings, American Mathematical Society, Colloquium Publications 37, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1037-8, MR 0081264, Zbl 0073.02002 • Kaplansky, Irving (1972), Fields and Rings, Chicago Lectures in Mathematics (2nd ed.), University Of Chicago Press, pp. 124–125, ISBN 0-226-42451-0, Zbl 1001.16500

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Chapter 243

Schanuel’s lemma In mathematics, especially in the area of algebra known as module theory, Schanuel’s lemma, named after Stephen Schanuel, allows one to compare how far modules depart from being projective. It is useful in defining the Heller operator in the stable category, and in giving elementary descriptions of dimension shifting.

243.1 Statement Schanuel’s lemma is the following statement: If 0 → K → P → M → 0 and 0 → K' → P ' → M → 0 are short exact sequences of R-modules and P and P ' are projective, then K ⊕ P ' is isomorphic to K ' ⊕ P.

243.2 Proof Define the following submodule of P ⊕ P ', where φ : P → M and φ' : P ' → M:

X = {(p, q) ∈ P ⊕ P ′ : ϕ(p) = ϕ′ (q)}. The map π : X → P, where π is defined as the projection of the first coordinate of X into P, is surjective. Since φ' is surjective, for any p ∈ P, one may find a q ∈ P ' such that φ(p) = φ '(q). This gives (p,q) ∈ X with π (p,q) = p. Now examine the kernel of the map π :

ker π = {(0, q) : (0, q) ∈ X} = {(0, q) : ϕ′ (q) = 0} ∼ = ker ϕ′ ∼ = K ′. We may conclude that there is a short exact sequence

0 → K ′ → X → P → 0. Since P is projective this sequence splits, so X ≅ K ' ⊕ P . Similarly, we can write another map π : X → P ', and the same argument as above shows that there is another short exact sequence

0 → K → X → P ′ → 0, and so X ≅ P ' ⊕ K. Combining the two equivalences for X gives the desired result. 745

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243.3 Long exact sequences The above argument may also be generalized to long exact sequences.[1]

243.4 Origins Stephen Schanuel discovered the argument in Irving Kaplansky's homological algebra course at the University of Chicago in Autumn of 1958. Kaplansky writes: Early in the course I formed a one-step projective resolution of a module, and remarked that if the kernel was projective in one resolution it was projective in all. I added that, although the statement was so simple and straightforward, it would be a while before we proved it. Steve Schanuel spoke up and told me and the class that it was quite easy, and thereupon sketched what has come to be known as “Schanuel’s lemma.” [2]

243.5 Notes [1] Lam, T.Y. (1999). Lectures on Modules and Rings. Springer. ISBN 0-387-98428-3. pgs. 165–167. [2] Kaplansky, Irving (1972). Fields and Rings. Chicago Lectures in Mathematics (2nd ed.). University Of Chicago Press. pp. 165–168. ISBN 0-226-42451-0. Zbl 1001.16500.

Chapter 244

Schreier domain In abstract algebra, a Schreier domain, named after Otto Schreier, is an integrally closed domain where every nonzero element is primal; i.e., whenever x divides yz, x can be written as x = x1 x2 so that x1 divides y and x2 divides z. An integral domain is said to be pre-Schreier if every nonzero element is primal. A GCD domain is an example of a Schreier domain. The term “Schreier domain” was introduced by P. M. Cohn in 1960s. The term “pre-Schreier domain” is due to Muhammad Zafrullah. In general, an irreducible element is primal if and only if it is a prime element. Consequently, in a Schreier domain, every irreducible is prime. In particular, an atomic Schreier domain is a unique factorization domain; this generalizes the fact that an atomic GCD domain is a UFD.

244.1 References • Cohn, P.M., Bezout rings and their subrings, 1967. • Zafrullah, Muhammad, On a property of pre-Schreier domains, 1987.

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Chapter 245

Semi-local ring For the older meaning of a Noetherian ring with a topology defined by an ideal in the Jacobson radical, see Zariski ring. In mathematics, a semi-local ring is a ring for which R/J(R) is a semisimple ring, where J(R) is the Jacobson radical of R. (Lam 2001, §20)(Mikhalev 2002, C.7) The above definition is satisfied if R has a finite number of maximal right ideals (and finite number of maximal left ideals). When R is a commutative ring, the converse implication is also true, and so the definition of semi-local for commutative rings is often taken to be “having finitely many maximal ideals". Some literature refers to a commutative semi-local ring in general as a quasi-semi-local ring, using semi-local ring to refer to a Noetherian ring with finitely many maximal ideals. A semi-local ring is thus more general than a local ring, which has only one maximal (right/left/two-sided) ideal.

245.1 Examples • Any right or left Artinian ring, any serial ring, and any semiperfect ring is semi-local. • The quotient Z/mZ is a semi-local ring. In particular, if m is a prime power, then Z/mZ is a local ring. ⊕n • A finite direct sum of fields i=1 Fi is a semi-local ring. • In the case of commutative rings with unity, this example is prototypical in the following sense: the Chinese remainder theorem shows that for a semi-local commutative ring R with unit and maximal ideals m1 , ..., mn ⊕n mi ∼ = i=1 R/mi . (The map is the natural projection). The right hand side is a direct sum of fields. Here we note that ∩ᵢ mᵢ=J(R), and we see that R/J(R) is indeed a semisimple ring. R/

∩n

i=1

• The classical ring of quotients for any commutative Noetherian ring is a semilocal ring. • The endomorphism ring of an Artinian module is a semilocal ring. • Semi-local rings occur for example in algebraic geometry when a (commutative) ring R is localized with respect to the multiplicatively closed subset S = ∩ (R \ pi), where the pi are finitely many prime ideals.

245.2 Textbooks • Lam, T. Y. (2001), “7”, A first course in noncommutative rings, Graduate Texts in Mathematics 131 (2 ed.), New York: Springer-Verlag, pp. xx+385, ISBN 0-387-95183-0, MR 1838439 (2002c:16001)} • Mikhalev, Alexander V.; Pilz, Günter F., eds. (2002), The concise handbook of algebra, Dordrecht: Kluwer Academic Publishers, pp. xvi+618, ISBN 0-7923-7072-4, MR 1966155 (2004c:00001)

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Chapter 246

Semi-orthogonal matrix In linear algebra, a semi-orthogonal matrix is a non-square matrix with real entries where: if the number of columns exceeds the number of rows, then the rows are orthonormal vectors; but if the number of rows exceeds the number of columns, then the columns are orthonormal vectors. Equivalently, a non-square matrix A is semi-orthogonal if either AT A = I or AAT = I.

[1]

In the following, consider the case where A is an m × n matrix for m > n. Then

AT A = In , which implies the isometry property ∥Ax∥2 = ∥x∥2 for all x in Rn . [ ] 1 For example, is a semi-orthogonal matrix. 0 A semi-orthogonal matrix A is semi-unitary (either A† A = I or AA† = I) and either left-invertible or right-invertible (left-invertible if it has more rows than columns, otherwise right invertible). As a linear transformation applied from the left, a semi-orthogonal matrix with more rows than columns preserves the dot product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation or reflection.

246.1 References [1] Abadir, K.M., Magnus, J.R. (2005). Matrix Algebra. Cambridge University Press.

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Chapter 247

Semi-simplicity This article is about mathematical use. For the philosophical reduction thinking, see Reduction (philosophy). In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of simple objects, and simple objects are those which do not contain non-trivial sub-objects. The precise definitions of these words depends on the context. For example, if G is a finite group, then a finite-dimensional representation V over a field is said to be simple if the only subrepresentations it contains are either {0} or V (these are also called irreducible representations). Then Maschke’s theorem says that any finite-dimensional representation is a direct sum of simple representations (provided the characteristic does not divide the order of the group). So, in this case, every representation of a finite group is semisimple. Especially in algebra and representation theory, “semi-simplicity” is also called complete reducibility. For example, Weyl’s theorem on complete reducibility says a finite-dimensional representation of a semisimple compact Lie group is semisimple. A square matrix (in other words a linear operator T : V → V with V finite dimensional vector space) is said to be simple if the only subspaces which are invariant under T are {0} and V. If the field is algebraically closed (such as the complex numbers), then the only simple matrices are of size 1 by 1. A semi-simple matrix is one that is similar to a direct sum of simple matrices; if the field is algebraically closed, this is the same as being diagonalizable. These notions of semi-simplicity can be unified using the language of semi-simple modules, and generalized to semisimple categories.

247.1 Introductory example of vector spaces If one considers all vector spaces (over a field, such as the real numbers), the simple vector spaces are those which contain no proper subspaces. Therefore, the one-dimensional vector spaces are the simple ones. So it is a basic result of linear algebra that any finite-dimensional vector space is the direct sum of simple vector spaces; in other words, all finite-dimensional vector spaces are semi-simple.

247.2 Semi-simple modules and rings Further information: Semisimple module and Semisimple ring For a fixed ring R, an R-module M is simple, if it has no submodules other than 0 and M. The module M is semi-simple if it is the direct sum of simple modules. Finally, R is called a semi-simple ring if it is semi-simple as an R-module. As it turns out, this is equivalent to requiring that any finitely generated R-module M is semi-simple.[1] Examples of semi-simple rings include fields and, more generally, finite direct products of fields. For a finite group G Maschke’s theorem asserts that the group ring R[G] over some ring R is semi-simple if and only if R is semisimple and |G| is invertible in R. Since the theory of modules of R[G] is the same as the representation theory of G 750

247.3. SEMI-SIMPLE MATRICES

751

on R-modules, this fact is an important dichotomy, which causes modular representation theory, i.e., the case when |G| does divide the characteristic of R to be more difficult than the case when |G| does not divide the characteristic, in particular if R is a field of characteristic zero. By the Artin–Wedderburn theorem, a unital Artinian ring R is semisimple if and only if it is (isomorphic to) Mn (D1 ) × Mn (D2 ) × · · · × Mn (Dr ) , where each Di is a division ring and Mn (D) is the ring of n-by-n matrices with entries in D. As indicated above, the theory of semi-simple rings is much more easy than the one of general rings. For example, any short exact sequence 0 → M ′ → M → M ′′ → 0 of modules over a semi-simple ring must split, i.e., M ∼ = M ′ ⊕ M ′′ . From the point of view of homological algebra, this means that there are no non-trivial extensions. The ring Z of integers is not semi-simple: Z is not the direct sum of nZ and Z/n.

247.3 Semi-simple matrices A matrix or, equivalently, a linear operator T on a finite-dimensional vector space V is called semi-simple if every T-invariant subspace has a complementary T-invariant subspace.[2][3] This is equivalent to the minimal polynomial of T being square-free. For vector spaces over an algebraically closed field F, semi-simplicity of a matrix is equivalent to diagonalizability.[2] This is because such an operator always has an eigenvector; if it is, in addition, semi-simple, then it has a complementary invariant hyperplane, which itself has an eigenvector, and thus by induction is diagonalizable. Conversely, diagonalizable operators are easily seen to be semi-simple, as invariant subspaces are direct sums of eigenspaces, and any basis for this space can be extended to an eigenbasis. Actually this notion of semi-simplicity is a special case of the one of rings: T is semi-simple if and only if the subalgebra F [T ] ⊂ EndF (V ) generated by the powers (i.e., iterations) of T inside the ring of endomorphisms of V is semi-simple.

247.4 Semi-simple categories Many of the above notions of semi-simplicity are recovered by the concept of a semi-simple category C. Briefly, a category is a collection of objects and maps between such objects, the idea being that the maps between the objects preserve some structure inherent in these objects. For example, R-modules and R-linear maps between them form a category, for any ring R. An abelian category[4] C is called semi-simple if there is a collection of simple objects Xα ∈ C , i.e., ones which have no subobject other than the zero object 0 and Xα itself, such that any object X is the direct sum (i.e., coproduct or, equivalently, product) of simple objects. With this terminology, a ring R is semi-simple if and only if the category of finitely generated R-modules is semisimple. An example from Hodge theory is the category of polarizable pure Hodge structures, i.e., pure Hodge structures equipped with a suitable positive definite bilinear. The presence of this so-called polarization causes the category of polarizable Hodge structures to be semi-simple.[5] Another example from algebraic geometry is the category of pure motives of smooth projective varieties over a field k Mot(k)∼ modulo an adequate equivalence relation ∼ . As was conjectured by Grothendieck and shown by Jannsen, this category is semi-simple if and only if the equivalence relation is numerical equivalence.[6] This fact is a conceptual cornerstone in the theory of motives.

247.5 Semi-simplicity in representation theory One can ask whether the category of (say, finite-dimensional) representations of a group G is semisimple or not (in such a category, irreducible representations are precisely simple objects). For example, the category is semisimple if G is a semisimple compact Lie group (Weyl’s theorem on complete reducibility). See also: fusion category (which is semisimple).

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247.6 See also • A semisimple Lie algebra is a Lie algebra that is a direct sum of simple Lie algebras. • A semisimple algebraic group is a linear algebraic group whose radical of the identity component is trivial. • Semisimple algebra

247.7 References [1]

• Lam, Tsit-Yuen (2001). A first course in noncommutative rings. Graduate texts in mathematics 131 (2 ed.). Springer. ISBN 0-387-95183-0.

[2] Lam (2001), p. 39 [3] Hoffman, Kenneth; Kunze, Ray (1971). “Semi-Simple operators”. Linear algebra (2nd ed.). Englewood Cliffs, N.J.: Prentice-Hall, Inc. MR 0276251. [4] More generally, the same definition of semi-simplicity works for pseudo-abelian additive categories. See for example Yves André, Bruno Kahn: Nilpotence, radicaux et structures monoïdales. With an appendix by Peter O'Sullivan. Rend. Sem. Mat. Univ. Padova 108 (2002), 107–291. http://arxiv.org/abs/math/0203273. [5] Peters, Chris A. M.; Steenbrink, Joseph H. M. Mixed Hodge structures. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 52. Springer-Verlag, Berlin, 2008. xiv+470 pp. ISBN 978-3-540-77015-2; see Corollary 2.12 [6] Uwe Jannsen: Motives, numerical equivalence, and semi-simplicity, Invent. math. 107, 447~452 (1992)

247.8 External links • http://mathoverflow.net/questions/245/are-abelian-nondegenerate-tensor-categories-semisimple • http://ncatlab.org/nlab/show/semisimple+category

Chapter 248

Seminormal ring In algebra, a seminormal ring is a commutative reduced ring in which, whenever x, y satisfy x3 = y 2 , there is s with s2 = x and s3 = y . This definition was given by Swan (1980) as a simplification of the original definition of Traverso (1970). A basic example is an integrally closed domain. A semigroup is said to be seminormal if its semigroup algebra is seminormal.

248.1 References • Swan, Richard G. (1980), “On seminormality”, Journal of Algebra 67 (1): 210–229, doi:10.1016/00218693(80)90318-X, ISSN 0021-8693, MR 595029 • Traverso, Carlo (1970), “Seminormality and Picard group”, Ann. Scuola Norm. Sup. Pisa (3) 24: 585–595, MR 0277542 • Vitulli, Marie A. (2011), “Weak normality and seminormality”, Commutative algebra---Noetherian and nonNoetherian perspectives, Berlin, New York: Springer-Verlag, pp. 441–480, doi:10.1007/978-1-4419-69903_17, MR 2762521 • Charles Weibel, The K-book: An introduction to algebraic K-theory

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Chapter 249

Semiprime ring

A Hasse diagram of a portion of the lattice of ideals of the integers Z. The purple and green nodes indicate semiprime ideals. The purple nodes are prime ideals, and the purple and blue nodes are primary ideals.

In ring theory, a branch of mathematics, semiprime ideals and semiprime rings are generalizations of prime ideals and prime rings. In commutative algebra, semiprime ideals are also called radical ideals. For example, in the ring of integers, the semiprime ideals are the zero ideal, along with those ideals of the form nZ where n is a square-free integer. So, 30Z is a semiprime ideal of the integers, but 12Z is not. The class of semiprime rings includes semiprimitive rings, prime rings and reduced rings. Most definitions and assertions in this article appear in (Lam 1999) and (Lam 2001).

249.1 Definitions For a commutative ring R, a proper ideal A is a semiprime ideal if A satisfies either of the following equivalent conditions: • If xk is in A for some positive integer k and element x of R, then x is in A. • If y is in R but not in A, all positive integer powers of y are not in A. The latter condition that the complement is “closed under powers” is analogous to the fact that complements of prime ideals are closed under multiplication. 754

249.2. GENERAL PROPERTIES OF SEMIPRIME IDEALS

755

As with prime ideals, this is extended to noncommutative rings “ideal-wise”. The following conditions are equivalent definitions for a semiprime ideal A in a ring R: • For any ideal J of R, if J k ⊆A for a positive natural number k, then J⊆A. • For any right ideal J of R, if J k ⊆A for a positive natural number k, then J⊆A. • For any left ideal J of R, if J k ⊆A for a positive natural number k, then J⊆A. • For any x in R, if xRx⊆A, then x is in A. Here again, there is a noncommutative analogue of prime ideals as complements of m-systems. A nonempty subset S of a ring R is called an n-system if for any s in S, there exists an r in R such that srs is in S. With this notion, an additional equivalent point may be added to the above list: • R\A is an n-system. The ring R is called a semiprime ring if the zero ideal is a semiprime ideal. In the commutative case, this is equivalent to R being a reduced ring, since R has no nonzero nilpotent elements. In the noncommutative case, the ring merely has no nonzero nilpotent right ideals. So while a reduced ring is always semiprime, the converse is not true.[1]

249.2 General properties of semiprime ideals To begin with, it is clear that prime ideals are semiprime, and that for commutative rings, a semiprime primary ideal is prime. While the intersection of prime ideals is not usually prime, it is a semiprime ideal. Shortly it will be shown that the converse is also true, that every semiprime ideal is the intersection of a family of prime ideals. For any ideal B in a ring R, we can form the following sets: ∩ √ B := {P ⊆ R | B ⊆ P, P a prime ideal} ⊆ {x ∈ R | xn ∈ B for some n ∈ N+ } √ The set B is the definition of the radical of B and is clearly a semiprime ideal containing B, and in fact is the smallest semiprime ideal containing B. The inclusion above is sometimes proper in the general case, but for commutative rings it becomes an equality. √ With this definition, an ideal A is semiprime if and only if A = A . At this point, it is also apparent that every semiprime ideal is in fact the intersection of a family of prime ideals. Moreover, this shows that the intersection of any two semiprime ideals is again semiprime. √ By definition R is semiprime if and only if {0} = {0} , that is, the intersection of all prime ideals is zero. This √ ideal {0} is also denoted by N il∗ (R) and also called Baer’s lower nilradical or the Baer-Mccoy radical or the prime radical of R.

249.3 Semiprime Goldie rings Main article: Goldie ring

249.4 References [1] The full ring of two-by-two matrices over a field is semiprime with nonzero nilpotent elements.

• Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294

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• Lam, T. Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics 131 (2 ed.), New York: Springer-Verlag, pp. xx+385, ISBN 0-387-95183-0, MR 1838439

249.5 External links • PlanetMath article on semiprime ideals

Chapter 250

Semiprimitive ring In algebra, a semiprimitive ring or Jacobson semisimple ring or J-semisimple ring is a ring whose Jacobson radical is zero. This is a type of ring more general than a semisimple ring, but where simple modules still provide enough information about the ring. Rings such as the ring of integers are semiprimitive, and an artinian semiprimitive ring is just a semisimple ring. Semiprimitive rings can be understood as subdirect products of primitive rings, which are described by the Jacobson density theorem.

250.1 Definition A ring is called semiprimitive or Jacobson semisimple if its Jacobson radical is the zero ideal. A ring is semiprimitive if and only if it has a faithful semisimple left module. The semiprimitive property is left-right symmetric, and so a ring is semiprimitive if and only if it has a faithful semisimple right module. A ring is semiprimitive if and only if it is a subdirect product of left primitive rings. A commutative ring is semiprimitive if and only if it is a subdirect product of fields, (Lam 1995, p. 137). A left artinian ring is semiprimitive if and only if it is semisimple, (Lam 2001, p. 54). Such rings are sometimes called semisimple Artinian, (Kelarev 2002, p. 13).

250.2 Examples • The ring of integers is semiprimitive, but not semisimple. • Every primitive ring is semiprimitive. • The product of two fields is semiprimitive but not primitive. • Every von Neumann regular ring is semiprimitive. Jacobson himself has defined a ring to be “semisimple” if and only if it is a subdirect product of simple rings, (Jacobson 1989, p. 203). However, this is a stricter notion, since the endomorphism ring of a countably infinite dimensional vector space is semiprimitive, but not a subdirect product of simple rings, (Lam 1995, p. 42).

250.3 References • Jacobson, Nathan (1989), Basic algebra II (2nd ed.), W. H. Freeman, ISBN 978-0-7167-1933-5 • Lam, Tsit-Yuen (1995), Exercises in classical ring theory, Problem Books in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94317-6, MR 1323431 757

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• Lam, Tsit-Yuen (2001), A First Course in Noncommutative Rings, Berlin, New York: Springer-Verlag, ISBN 978-0-387-95325-0 • Kelarev, Andrei V. (2002), Ring Constructions and Applications, World Scientific, ISBN 978-981-02-4745-4

Chapter 251

Semiring In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally[1] — this originated as a joke, suggesting that rigs are rings without negative elements, similar to using rng to mean a ring without a multiplicative identity.

251.1 Definition A semiring is a set R equipped with two binary operations + and ·, called addition and multiplication, such that:[2][3][4] 1. (R, +) is a commutative monoid with identity element 0: (a) (a + b) + c = a + (b + c) (b) 0 + a = a + 0 = a (c) a + b = b + a 2. (R, ·) is a monoid with identity element 1: (a) (a·b)·c = a·(b·c) (b) 1·a = a·1 = a 3. Multiplication left and right distributes over addition: (a) a·(b + c) = (a·b) + (a·c) (b) (a + b)·c = (a·c) + (b·c) 4. Multiplication by 0 annihilates R: (a) 0·a = a·0 = 0 This last axiom is omitted from the definition of a ring: it follows from the other ring axioms. Here it does not, and it is necessary to state it in the definition. The difference between rings and semirings, then, is that addition yields only a commutative monoid, not necessarily a commutative group. Specifically, elements in semirings do not necessarily have an inverse for the addition. The symbol · is usually omitted from the notation; that is, a·b is just written ab. Similarly, an order of operations is accepted, according to which · is applied before +; that is, a + bc is a + (bc). A commutative semiring is one whose multiplication is commutative.[5] An idempotent semiring is one whose addition is idempotent: a + a = a,[6] that is, (R, +, 0) is a join-semilattice with zero. There are some authors who prefer to leave out the requirement that a semiring have a 0 or 1. This makes the analogy between ring and semiring on the one hand and group and semigroup on the other hand work more smoothly. These authors often use rig for the concept defined here.[note 1] 759

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251.2 Examples 251.2.1

In general

• Any ring is also a semiring. • The set of all ideals of a given ring form a semiring under addition and multiplication of ideals. • Any unital quantale is an idempotent semiring, or dioid, under join and multiplication. • Any bounded, distributive lattice is a commutative, idempotent semiring under join and meet. • In particular, a Boolean algebra is such a semiring. A Boolean ring is also a semiring—indeed, a ring—but it is not idempotent under addition. A Boolean semiring is a semiring isomorphic to a subsemiring of a Boolean algebra.[7] • A normal skew lattice in a ring R is an idempotent semiring for the operations multiplication and nabla, where the latter operation is defined by a∇b = a + b + ba − aba − bab . • Any c-semiring is also a semiring, where addition is idempotent and defined over arbitrary sets.

251.2.2

Specific examples

• A motivating example of a semiring is the set of natural numbers N (including zero) under ordinary addition and multiplication. Likewise, the non-negative rational numbers and the non-negative real numbers form semirings. All these semirings are commutative.[7][8][9] • The extended natural numbers N∪{∞} with addition and multiplication extended (and 0⋅∞ = ∞).[8] • The square n-by-n matrices with non-negative entries form a (not necessarily commutative) semiring under ordinary addition and multiplication of matrices. More generally, this likewise applies to the square matrices whose entries are elements of any other given semiring S, and this new semiring of matrices is generally noncommutative even though S may be commutative.[7] • If A is a commutative monoid, the set End(A) of endomorphisms f:A→A form a semiring, where addition is pointwise addition and multiplication is function composition. The zero morphism and the identity are the respective neutral elements. If A is the additive monoid of natural numbers we obtain the semiring of natural numbers as End(A), and if A=S^n with S a semiring, we obtain (after associating each morphism to a matrix) the semiring of square n-by-n matrices with coefficients in S. • The Boolean semiring: the commutative semiring B formed by the two-element Boolean algebra and defined by 1+1=1:[3][8][9] this is idempotent[6] and is the simplest example of a semiring which is not a ring. • N[x], polynomials with natural number coefficients form a commutative semiring. In fact, this is the free commutative semiring on a single generator {x}. • Tropical semirings are variously defined. The max-plus semiring R ∪ {−∞}, is a commutative, idempotent semiring with max(a,b) serving as semiring addition (identity −∞) and ordinary addition (identity 0) serving as semiring multiplication. In an alternative formulation, the tropical semiring is R ∪ {∞}, and min replaces max as the addition operation.[10] A related version has R ∪ {±∞} as the underlying set.[3][11] • The set of cardinal numbers smaller than any given infinite cardinal form a semiring under cardinal addition and multiplication. The class of all cardinals of an inner model form a (class) semiring under (inner model) cardinal addition and multiplication. • The probability semiring of non-negative real numbers under the usual addition and multiplication.[3] • The log semiring on R ∪ ±∞ with addition given by x ⊕ y = − log(e−x + e−y ) , with multiplication +, zero element +∞ and unit element 0.[3]

251.3. SEMIRING THEORY

761

• The family of (isomorphism equivalence classes of) combinatorial classes (sets of countably many objects with non-negative integer sizes such that there are finitely many objects of each size) with the empty class as the zero object, the class consisting only of the empty set as the unit, disjoint union of classes as addition, and Cartesian product of classes as multiplication.[12] • The Łukasiewicz semiring: the closed interval [0, 1] with addition given by taking the maximum of the arguments (a+b=max(a,b)) and multiplication ab given by max(0, a+b−1) appears in multi-valued logic.[13] • The Viterbi semiring is also over the base set [0, 1] and addition by maximum, but with multiplication as the usual multiplication of reals; it appears in probabilistic parsing.[13] • Given a set U, the set of binary relations over U is a semiring with addition the union (of relations as sets) and multiplication the composition of relations. The semiring’s zero is the empty relation and its unit is the identity relation.[13] • Given an alphabet (finite set) Σ, the set of formal languages over Σ (subsets of Σ* ) is a semiring with product induced by string concatenation L1 ·L2 = {w1 w2 | w1 ∈ L1 , w2 ∈ L2 } and addition as the union of languages (i.e. simply union as sets). The zero of this semiring is the empty set (empty language) and the semiring’s unit is the language containing as its sole element the empty string.[13] • Generalising the previous example by viewing Σ∗ as the free monoid over Σ, take M to be any monoid; the power set P M of all subsets of M forms a semiring under set-theoretic union as addition and set-wise multiplication: U · V = {u · v : u ∈ U, v ∈ v} .[9]

251.3 Semiring theory Much of the theory of rings continues to make sense when applied to arbitrary semirings. In particular, one can generalise the theory of algebras over commutative rings directly to a theory of algebras over commutative semirings. Then a ring is simply an algebra over the commutative semiring Z of integers. Some mathematicians go so far as to say that semirings are really the more fundamental concept, and specialising to rings should be seen in the same light as specialising to, say, algebras over the complex numbers. Idempotent semirings are special to semiring theory as any ring which is idempotent under addition is trivial. One can define a partial order ≤ on an idempotent semiring by setting a ≤ b whenever a + b = b (or, equivalently, if there exists an x such that a + x = b). It is easy to see that 0 is the least element with respect to this order: 0 ≤ a for all a. Addition and multiplication respect the ordering in the sense that a ≤ b implies ac ≤ bc and ca ≤ cb and (a+c) ≤ (b+c).

251.4 Applications Semirings, especially the (max, +) and (min, +) semirings on the reals, are often used in performance evaluation on discrete event systems. The real numbers then are the “costs” or “arrival time"; the “max” operation corresponds to having to wait for all prerequisites of an events (thus taking the maximal time) while the “min” operation corresponds to being able to choose the best, less costly choice; and + corresponds to accumulation along the same path. The Floyd–Warshall algorithm for shortest paths can thus be reformulated as a computation over a (min, +) algebra. Similarly, the Viterbi algorithm for finding the most probable state sequence corresponding to an observation sequence in a Hidden Markov model can also be formulated as a computation over a (max, ×) algebra on probabilities. These dynamic programming algorithms rely on the distributive property of their associated semirings to compute quantities over a large (possibly exponential) number of terms more efficiently than enumerating each of them.

251.5 Complete and continuous semirings A complete semiring is a semiring for which the addition monoid is a complete monoid, meaning that it has an infinitary sum operation ΣI for any index set I and that the following (infinitary) distributive laws must hold:[11][13][14]

762 ∑ i∈I

CHAPTER 251. SEMIRING (a · ai ) = a · (

∑ i∈I

ai );

∑ i∈I

(ai · a) = (

∑ i∈I

ai ) · a.

Examples of complete semirings include the power set of a monoid under union; the matrix semiring over a complete semiring is complete.[15] A continuous semiring is similarly defined as one for which the addition monoid is a continuous monoid: that is, partially ordered with the least upper bounds property, and for which addition and multiplication respect order and suprema. The semiring N∪{∞} with usual addition, multiplication and order extended, is a continuous semiring.[16] Any continuous semiring is complete:[11] this may be taken as part of the definition.[15]

251.6 Star semirings A star semiring (sometimes spelled as starsemiring) is a semiring with an additional unary operator *,[6][13][17][18] satisfying

a∗ = 1 + aa∗ = 1 + a∗ a. Examples of star semirings include: ∪ • the (aforementioned) semiring of binary relations over some base set U in which R∗ = n≥0 Rn for all R ⊆ U × U . This star operation is actually the reflexive and transitive closure of R (i.e. the smallest reflexive and transitive binary relation over U containing R.).[13] • the semiring of formal languages is also a complete star semiring, with the star operation coinciding with the Kleene star (for sets/languages).[13] • The set of non-negative extended reals, [0, ∞], together with the usual addition and multiplication of reals is a complete star semiring with the star operation given by a∗ = 1/(1 − a) for 0 ≤ a < 1 (i.e. the geometric series) and a∗ = ∞ for a ≥ 1.[13] Further examples:[13] • The Boolean semiring with 0∗ = 1∗ = 1; • The semiring on N ∪ {∞}, with extended addition and multiplication, and 0∗ = 1, a∗ = ∞ for a ≥ 1. A Kleene algebra is a star semiring with idempotent addition: they are important in the theory of formal languages and regular expressions. A Conway semiring is a star semiring satisfying the sum-star and the product-star equations:[6][19]

(a + b)∗ = (a∗ b)∗ a∗ , (ab)∗ = 1 + a(ba)∗ b. The first three examples above are also Conway semirings.[13] An iteration semiring is a Conway semiring satisfying the Conway group axioms,[6] associated by John Conway to groups in star-semirings.[20]

251.6.1

Complete star semirings

We define a notion of complete star semiring in which the star operator behaves more like the usual Kleene star: for a complete semiring we use the infinitary sum operator to give the usual definition of the Kleene star:[13] ∑ a∗ = j≥0 aj where a0 = 1 and aj+1 = a · aj = aj · a for j ≥ 0 Examples of complete star semirings include the first three classes of examples in the previous section: the binary relations semiring; the formal languages semiring and the extended non-negative reals.[13]

251.7. FURTHER GENERALIZATIONS

763

In general, every complete star semiring is also a Conway semiring,[21] but the converse does not hold. An example of Conway semiring that is not complete is the set of extended non-negative rational numbers ({x ∈ Q | x ≥ 0} ∪ {∞}) with the usual addition and multplication (this is a modification of the example with extended non-negative reals given in this section by eliminating irrational numbers).[13]

251.7 Further generalizations A near-ring does not require addition to be commutative, nor does it require right-distributivity. Just as cardinal numbers form a (class) semiring, so do ordinal numbers form a near-ring, when the standard ordinal addition and multiplication are taken into account. However, the class of ordinals can be turned into a semiring by considering the so-called natural (or Hessenberg) operations instead. In category theory, a 2-rig is a category with functorial operations analogous to those of a rig. That the cardinal numbers form a rig can be categorified to say that the category of sets (or more generally, any topos) is a 2-rig.

251.8 Semiring of sets A semiring (of sets)[22] is a non-empty collection S of sets such that 1. ∅ ∈ S 2. If E ∈ S and F ∈ S then E ∩ F ∈ S . 3. If E ∈ S and ∪ F ∈ S then there exists a finite number of mutually disjoint sets Ci ∈ S for i = 1, . . . , n such n that E \ F = i=1 Ci . Such semirings are used in measure theory. An example of a semiring of sets is the collection of half-open, halfclosed real intervals [a, b) ⊂ R .

251.9 Terminology The term dioid (for “double monoid”) was used by Kuntzman in 1972 to denote what is now termed semiring.[23] The use to mean idempotent subgroup was introduced by Baccelli et al. in 1992.[24]

251.10 See also • Ring of sets • Valuation algebra

251.11 Notes [1] For an example see the definition of rig on Proofwiki.org

251.12 Bibliography [1] Głazek (2002) p.7 [2] Berstel & Perrin (1985), p. 26 [3] Lothaire (2005) p.211

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[4] Sakarovitch (2009) pp.27–28 [5] Lothaire (2005) p.212 [6] Ésik, Zoltán (2008). “Iteration semirings”. In Ito, Masami. Developments in language theory. 12th international conference, DLT 2008, Kyoto, Japan, September 16–19, 2008. Proceedings. Lecture Notes in Computer Science 5257. Berlin: Springer-Verlag. pp. 1–20. doi:10.1007/978-3-540-85780-8_1. ISBN 978-3-540-85779-2. Zbl 1161.68598. [7] Guterman, Alexander E. (2008). “Rank and determinant functions for matrices over semirings”. In Young, Nicholas; Choi, Yemon. Surveys in Contemporary Mathematics. London Mathematical Society Lecture Note Series 347. Cambridge University Press. pp. 1–33. ISBN 0-521-70564-9. ISSN 0076-0552. Zbl 1181.16042. [8] Sakarovitch (2009) p.28 [9] Berstel & Reutenauer (2011) p.4 [10] Speyer, David; Sturmfels, Bernd (2009) [2004]. “Tropical Mathematics”. Math. Mag. 82 (3): 163–173. arXiv:math/0408099. doi:10.4169/193009809x468760. Zbl 1227.14051. [11] Kuich, Werner (2011). “Algebraic systems and pushdown automata”. In Kuich, Werner. Algebraic foundations in computer science. Essays dedicated to Symeon Bozapalidis on the occasion of his retirement. Lecture Notes in Computer Science 7020. Berlin: Springer-Verlag. pp. 228–256. ISBN 978-3-642-24896-2. Zbl 1251.68135. [12] Bard, Gregory V. (2009), Algebraic Cryptanalysis, Springer, Section 4.2.1, “Combinatorial Classes”, ff., pp. 30–34, ISBN 9780387887579. [13] Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. Handbook of Weighted Automata, 3–28. doi:10.1007/9783-642-01492-5_1, pp. 7-10 [14] Kuich, Werner (1990). "ω-continuous semirings, algebraic systems and pushdown automata”. In Paterson, Michael S. Automata, Languages and Programming: 17th International Colloquium, Warwick University, England, July 16-20, 1990, Proceedings. Lecture Notes in Computer Science 443. Springer-Verlag. pp. 103–110. ISBN 3-540-52826-1. [15] Sakaraovich (2009) p.471 [16] Ésik, Zoltán; Leiß, Hans (2002). “Greibach normal form in algebraically complete semirings”. In Bradfield, Julian. Computer science logic. 16th international workshop, CSL 2002, 11th annual conference of the EACSL, Edinburgh, Scotland, September 22-25, 2002. Proceedings. Lecture Notes in Computer Science 2471. Berlin: Springer-Verlag. pp. 135–150. Zbl 1020.68056. [17] Lehmann, Daniel J. “Algebraic structures for transitive closure.” Theoretical Computer Science 4, no. 1 (1977): 59-76. [18] Berstel & Reutenauer (2011) p.27 [19] Ésik, Zoltán; Kuich, Werner (2004). “Equational axioms for a theory of automata”. In Martín-Vide, Carlos. Formal languages and applications. Studies in Fuzziness and Soft Computing 148. Berlin: Springer-Verlag. pp. 183–196. ISBN 3-540-20907-7. Zbl 1088.68117. [20] Conway, J.H. (1971). Regular algebra and finite machines. London: Chapman and Hall. ISBN 0-412-10620-5. Zbl 0231.94041. [21] Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. Handbook of Weighted Automata, 3–28. doi:10.1007/9783-642-01492-5_1, Theorem 3.4 p. 15 [22] Noel Vaillant, Caratheodory’s Extension, on probability.net. [23] Kuntzmann, J. (1972). Théorie des réseaux (graphes) (in French). Paris: Dunod. Zbl 0239.05101. [24] Baccelli, François Louis; Olsder, Geert Jan; Quadrat, Jean-Pierre; Cohen, Guy (1992). Synchronization and linearity. An algebra for discrete event systems. Wiley Series on Probability and Mathematical Statistics. Chichester: Wiley. Zbl 0824.93003.

• François Baccelli, Guy Cohen, Geert Jan Olsder, Jean-Pierre Quadrat, Synchronization and Linearity (online version), Wiley, 1992, ISBN 0-471-93609-X • Golan, Jonathan S., Semirings and their applications. Updated and expanded version of The theory of semirings, with applications to mathematics and theoretical computer science (Longman Sci. Tech., Harlow, 1992, MR 1163371. Kluwer Academic Publishers, Dordrecht, 1999. xii+381 pp. ISBN 0-7923-5786-8 MR 1746739

251.13. FURTHER READING

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• Berstel, Jean; Perrin, Dominique (1985). Theory of codes. Pure and applied mathematics 117. Academic Press. ISBN 978-0-12-093420-1. Zbl 0587.68066. • Lothaire, M. (2005). Applied combinatorics on words. Encyclopedia of Mathematics and Its Applications 105. A collective work by Jean Berstel, Dominique Perrin, Maxime Crochemore, Eric Laporte, Mehryar Mohri, Nadia Pisanti, Marie-France Sagot, Gesine Reinert, Sophie Schbath, Michael Waterman, Philippe Jacquet, Wojciech Szpankowski, Dominique Poulalhon, Gilles Schaeffer, Roman Kolpakov, Gregory Koucherov, JeanPaul Allouche and Valérie Berthé. Cambridge: Cambridge University Press. ISBN 0-521-84802-4. Zbl 1133.68067. • Głazek, Kazimierz (2002). A guide to the literature on semirings and their applications in mathematics and information sciences. With complete bibliography. Dordrecht: Kluwer Academic. ISBN 1-4020-0717-5. Zbl 1072.16040. • Sakarovitch, Jacques (2009). Elements of automata theory. Translated from the French by Reuben Thomas. Cambridge: Cambridge University Press. ISBN 978-0-521-84425-3. Zbl 1188.68177. • Berstel, Jean; Reutenauer, Christophe (2011). Noncommutative rational series with applications. Encyclopedia of Mathematics and Its Applications 137. Cambridge: Cambridge University Press. ISBN 978-0-521-190220. Zbl 1250.68007.

251.13 Further reading • Golan, Jonathan S. (2003). Semirings and Affine Equations over Them. Springer Science & Business Media. ISBN 978-1-4020-1358-4. Zbl 1042.16038. • Gondran, Michel; Minoux, Michel (2008). Graphs, Dioids and Semirings: New Models and Algorithms. Operations Research/Computer Science Interfaces Series 41. Dordrecht: Springer Science & Business Media. ISBN 978-0-387-75450-5. Zbl 1201.16038. • Grillet, Mireille P. (1970). “Green’s relations in a semiring”. Port. Math. 29: 181–195. Zbl 0227.16029. • Gunawardena, Jeremy (1998). “An introduction to idempotency”. In Gunawardena, Jeremy. Idempotency. Based on a workshop, Bristol, UK, October 3–7, 1994 (PDF). Cambridge: Cambridge University Press. pp. 1–49. Zbl 0898.16032.

• Jipsen, P. (2004). “From semirings to residuated Kleene lattices”. Studia Logica 76 (2): 291–303. doi:10.1023/B:STUD.0000032 Zbl 1045.03049. • Steven Dolan (2013) Fun with Semirings, doi:10.1145/2500365.2500613

Chapter 252

Semisimple module In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as group rings of finite groups over fields of characteristic zero, are semisimple rings. An Artinian ring is initially understood via its largest semisimple quotient. The structure of Artinian semisimple rings is well understood by the Artin–Wedderburn theorem, which exhibits these rings as finite direct products of matrix rings.

252.1 Definition A module over a (not necessarily commutative) ring with unity is said to be semisimple (or completely reducible) if it is the direct sum of simple (irreducible) submodules. For a module M, the following are equivalent: 1. M is semisimple; i.e., a direct sum of irreducible modules. 2. M is the sum of its irreducible submodules. 3. Every submodule of M is a direct summand: for every submodule N of M, there is a complement P such that M = N ⊕ P. For 3 ⇒ 2 , the starting idea is to find an irreducible submodule by picking any nonzero x ∈ M and letting P be a maximal submodule such that x ∈ / P . It can be shown that the complement of P is irreducible.[1] The most basic example of a semisimple module is a module over a field; i.e., a vector space. On the other hand, the ring Z of integers is not a semisimple module over itself (because, for example, it is not an artinian ring.) Semisimple is stronger than completely decomposable, which is a direct sum of indecomposable submodules. Let A be an algebra over a field k. Then a left module M over A is said to be absolutely semisimple if, for any field extension F of k, F ⊗k M is a semisimple module over F ⊗k A .

252.2 Properties • If M is semisimple and N is a submodule, then N and M/N are also semisimple. • If each Mi is a semisimple module, then so is

⊕ i

Mi .

• A module M is finitely generated and semisimple if and only if it is Artinian and its radical is zero. 766

252.3. ENDOMORPHISM RINGS

767

252.3 Endomorphism rings • A semisimple module M over a ring R can also be thought of as a ring homomorphism from R into the ring of abelian group endomorphisms of M. The image of this homomorphism is a semiprimitive ring, and every semiprimitive ring is isomorphic to such an image. • The endomorphism ring of a semisimple module is not only semiprimitive, but also von Neumann regular, (Lam 2001, p. 62).

252.4 Semisimple rings A ring is said to be (left)-semisimple if it is semisimple as a left module over itself. Surprisingly, a left-semisimple ring is also right-semisimple and vice versa. The left/right distinction is therefore unnecessary, and one can speak of semisimple rings without ambiguity. A semisimple ring may be characterized in terms of homological algebra: namely, a ring R is semisimple if and only if any short exact sequence of left (or right) R-modules splits. In particular, any module over a semisimple ring is injective and projective. Since “projective” implies “flat”, a semisimple ring is a von Neumann regular ring. Semisimple rings are of particular interest to algebraists. For example, if the base ring R is semisimple, then all Rmodules would automatically be semisimple. Furthermore, every simple (left) R-module is isomorphic to a minimal left ideal of R, that is, R is a left Kasch ring. Semisimple rings are both Artinian and Noetherian. From the above properties, a ring is semisimple if and only if it is Artinian and its Jacobson radical is zero. If an Artinian semisimple ring contains a field, it is called a semisimple algebra.

252.4.1

Examples

• A commutative semisimple ring is a finite direct product of fields. A commutative ring is semisimple if and only if it is artinian and reduced.[2] • If k is a field and G is a finite group of order n, then the group ring k[G] is semisimple if and only if the characteristic of k does not divide n. This is Maschke’s theorem, an important result in group representation theory. • By the Artin–Wedderburn theorem, a unital Artinian ring R is semisimple if and only if it is (isomorphic to) Mn1 (D1 ) × Mn2 (D2 ) × · · · × Mnr (Dr ) , where each Di is a division ring and each ni is a positive integer, and Mn (D) denotes the ring of n-by-n matrices with entries in D. • An example of a semisimple non-unital ring is M∞ (K) , the row-finite, column-finite, infinite matrices over a field K.

252.4.2

Simple rings

Main article: simple ring One should beware that despite the terminology, not all simple rings are semisimple. The problem is that the ring may be “too big”, that is, not (left/right) Artinian. In fact, if R is a simple ring with a minimal left/right ideal, then R is semisimple. Classic examples of simple, but not semisimple, rings are the Weyl algebras, such as Q/(xy-yx−1), which is a simple noncommutative domain. These and many other nice examples are discussed in more detail in several noncommutative ring theory texts, including chapter 3 of Lam’s text, in which they are described as nonartinian simple rings. The module theory for the Weyl algebras is well studied and differs significantly from that of semisimple rings.

768

252.4.3

CHAPTER 252. SEMISIMPLE MODULE

Jacobson semisimple

Main article: semiprimitive ring A ring is called Jacobson semisimple (or J-semisimple or semiprimitive) if the intersection of the maximal left ideals is zero, that is, if the Jacobson radical is zero. Every ring that is semisimple as a module over itself has zero Jacobson radical, but not every ring with zero Jacobson radical is semisimple as a module over itself. A J-semisimple ring is semisimple if and only if it is an artinian ring, so semisimple rings are often called artinian semisimple rings to avoid confusion. For example the ring of integers, Z, is J-semisimple, but not artinian semisimple.

252.5 See also • Socle • semisimple algebra

252.6 References 252.6.1

Notes

[1] Nathan Jacobson, Basic Algebra II (Second Edition), p.120 [2] Bourbaki, VIII, pg. 133.

252.6.2

Textbooks

• Bourbaki, Algèbre • Jacobson, Nathan (1989), Basic algebra II (2nd ed.), W. H. Freeman, ISBN 978-0-7167-1933-5 • Lam, Tsit-Yuen (2001), A First Course in Noncommutative Rings (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-95325-0, MR 1838439 • R.S. Pierce. Associative Algebras. Graduate Texts in Mathematics vol 88.

Chapter 253

Serial module “Chain ring” redirects here. For the bicycle part, see Chainring. In abstract algebra, a uniserial module M is a module over a ring R, whose submodules are totally ordered by inclusion. This means simply that for any two submodules N 1 and N 2 of M, either N1 ⊆ N2 or N2 ⊆ N1 . A module is called a serial module if it is a direct sum of uniserial modules. A ring R is called a right uniserial ring if it is uniserial as a right module over itself, and likewise called a right serial ring if it is a right serial module over itself. Left uniserial and left serial rings are defined in an analogous way, and are in general distinct from their right counterparts. An easy motivational example is the quotient ring Z/nZ for any integer n > 1 . This ring is always serial, and is uniserial when n is a prime power. The term uniserial has been used differently from the above definition: for clarification see this section. A partial alphabetical list of important contributors to the theory of serial rings includes the mathematicians Keizo Asano, I. S. Cohen, P.M. Cohn, Yu. Drozd, D. Eisenbud, A. Facchini, A.W. Goldie, Phillip Griffith, I. Kaplansky, V.V Kirichenko, G. Köthe, H. Kuppisch, I. Murase, T. Nakayama, P. Příhoda, G. Puninski, and R. Warfield. References for each author can be found in (Puninski 2001) and (Hazewinkel 2004). Following the common ring theoretic convention, if a left/right dependent condition is given without mention of a side (for example, uniserial, serial, Artinian, Noetherian) then it is assumed the condition holds on both the left and right. Unless otherwise specified, each ring in this article is a ring with unity, and each module is unital.

253.1 Properties of uniserial and serial rings and modules It is immediate that in a uniserial R-module M, all submodules except M and 0 are simultaneously essential and superfluous. If M has a maximal submodule, then M is a local module. M is also clearly a uniform module and thus is directly indecomposable. It is also easy to see that every finitely generated submodule of M can be generated by a single element, and so M is a Bézout module. It is known that the endomorphism ring EndR(M) is a semilocal ring which is very close to a local ring in the sense that EndR(M) has at most two maximal right ideals. If M is required to be Artinian or Noetherian, then EndR(M) is a local ring. Since rings with unity always have a maximal right ideal, a right uniserial ring is necessarily local. As noted before, a finitely generated right ideal can be generated by a single element, and so right uniserial rings are right Bézout rings. A right serial ring R necessarily factors in the form R = ⊕ni=1 ei R where each eᵢ is an idempotent element and eᵢR is a local, uniserial module. This indicates that R is also a semiperfect ring, which is a stronger condition than being a semilocal ring. Köthe showed that the modules of Artinian principal ideal rings (which are a special case of serial rings) are direct sums of cyclic submodules. Later, Cohen and Kaplansky determined that a commutative ring R has this property for its modules if and only if R is an Artinian principal ideal ring. Nakayama showed that Artinian serial rings have this property on their modules, and that the converse is not true 769

770

CHAPTER 253. SERIAL MODULE

The most general result, perhaps, on the modules of a serial ring is attributed to Drozd and Warfield: it states that every finitely presented module over a serial ring is a direct sum of cyclic uniserial submodules (and hence is serial). If additionally the ring is assumed to be Noetherian, the finitely presented and finitely generated modules coincide, and so all finitely generated modules are serial. Being right serial is preserved under direct products of rings and modules, and preserved under quotients of rings. Being uniserial is preserved for quotients of rings and modules, but never for products. A direct summand of a serial module is not necessarily serial, as was proved by Puninski, but direct summands of finite direct sums of uniserial modules are serial modules (Příhoda 2004). It has been verified that Jacobson’s conjecture holds in Noetherian serial rings.(Chatters & Hajarnavis 1980)

253.2 Examples Any simple module is trivially uniserial, and likewise semisimple modules are serial modules. Many examples of serial rings can be gleaned from the structure sections above. Every valuation ring is a uniserial ring, and all Artinian principal ideal rings are serial rings, as is illustrated by semisimple rings. More exotic examples include the upper triangular matrices over a division ring T (D), and the group ring F[G] for some finite field of prime characteristic p and group G having a cyclic normal p-Sylow subgroup.

253.3 Structure This section will deal mainly with Noetherian serial rings and their subclass, Artinian serial rings. In general, rings are first broken down into indecomposable rings. Once the structure of these rings are known, the decomposable rings are direct products of the indecomposable ones. Also, for semiperfect rings such as serial rings, the basic ring is Morita equivalent to the original ring. Thus if R is a serial ring with basic ring B, and the structure of B is known, the theory of Morita equivalence gives that R ∼ = EndB (P ) where P is some finitely generated progenerator B. This is why the results are phrased in terms of indecomposable, basic rings. In 1975, Kirichenko and Warfield independently and simultaneously published analyses of the structure of Noetherian, non-Artinian serial rings. The results were the same however the methods they used were very different from each other. The study of hereditary, Noetherian, prime rings, as well as quivers defined on serial rings were important tools. The core result states that a right Noetherian, non-Artinian, basic, indecomposable serial ring can be described as a type of matrix ring over a Noetherian, uniserial domain V, whose Jacobson radical J(V) is nonzero. This matrix ring is a subring of M (V) for some n, and consists of matrices with entries from V on and above the diagonal, and entries from J(V) below. Artinian serial ring structure is classified in cases depending on the quiver structure. It turns out that the quiver structure for a basic, indecomposable, Artinian serial ring is always a circle or a line. In the case of the line quiver, the ring is isomorphic to the upper triangular matrices over a division ring (note the similarity to the structure of Noetherian serial rings in the preceding paragraph). A complete description of structure in the case of a circle quiver is beyond the scope of this article, but the complete description can be found in (Puninski 2001). To paraphrase the result as it appears there: A basic Artinian serial ring whose quiver is a circle is a homomorphic image of a “blow-up” of a basic, indecomposable, serial quasi-Frobenius ring.

253.4 A decomposition uniqueness property Two modules U and V are said to have the same monogeny class, denoted [U] =[V] , if there exists a monomorphism U → V and a monomorphism V → U . The dual notion can be defined: the modules are said to have the same epigeny class, denoted [U ]e = [V ]e , if there exists an epimorphism U → V and an epimorphism V → U . The following weak form of the Krull-Schmidt theorem holds. Let U 1 ,... U , V 1 , ..., V be n+t non-zero uniserial right modules over a ring R. Then the direct sums U1 ⊕ · · · ⊕ Un and V1 ⊕ · · · ⊕ Vt are isomorphic R-modules if and only if n=t and there exist two permutations σ and τ of 1,2,...,n such that [Ui ]m = [Vσ(i) ]m and [Ui ]e = [Vτ (i) ]e for every i=1,2,..., n.

253.5. NOTES ON ALTERNATE, SIMILAR AND RELATED TERMS

771

This result, due to Facchini, has been extended to infinite direct sums of uniserial modules by Příhoda in 2006. This extension involves the so-called quasismall uniserial modules. These modules were defined by Nguyen Viet Dung and Facchini, and their existence was proved by Puninski. The weak form of the Krull-Schmidt Theorem holds not only for uniserial modules, but also for several other classes of modules (biuniform modules, cyclically presented modules over serial rings, kernels of morphisms between indecomposable injective modules, couniformly presented modules.)

253.5 Notes on alternate, similar and related terms Right uniserial rings can also be referred to as right chain rings (Faith 1999) or right valuation rings. This latter term alludes to valuation rings, which are by definition commutative, uniserial domains. By the same token, uniserial modules have been called chain modules, and serial modules semichain modules. The notion of a catenary ring has “chain” as its namesake, but it is in general not related to chain rings. In the 1930s, Gottfried Köthe and Keizo Asano introduced the term Einreihig (literally “one-series”) during investigations of rings over which all modules are direct sums of cyclic submodules (Köthe 1935). For this reason, uniserial was used to mean “Artinian principal ideal ring” even as recently as the 1970s. Köthe’s paper also required a uniserial ring to have a unique composition series, which not only forces the right and left ideals to be linearly ordered, but also requires that there be only finitely many ideals in the chains of left and right ideals. Because of this historical precedent, some authors include the Artinian condition or finite composition length condition in their definitions of uniserial modules and rings. Expanding on Köthe’s work, Tadashi Nakayama used the term generalized uniserial ring (Nakayama 1941) to refer to an Artinian serial ring. Nakayama showed that all modules over such rings are serial. Artinian serial rings are sometimes called Nakayama algebras, and they have a well-developed module theory. Warfield used the term homogeneously serial module for a serial module with the additional property that for any two finitely generated submodules A and B, A/J(A) ∼ = B/J(B) where J(-) denotes the Jacobson radical of the module (Warfield 1975). In a module with finite composition length, this has the effect of forcing the composition factors to be isomorphic, hence the “homogeneous” adjective. It turns out that a serial ring R is a finite direct sum of homogeneously serial right ideals if and only if R is isomorphic to a full nxn matrix ring over a local serial ring. Such rings are also known as primary decomposable serial rings (Faith 1976)(Hazewinkel, Gubareni & Kirichenko 2004).

253.6 Textbooks • Frank W. Anderson and Kent R. Fuller (1992), Rings and Categories of Modules, Springer, pp. 347–349, ISBN 0-387-97845-3 • Chatters, A. W.; Hajarnavis, C.R. (1980), Rings with chain conditions, Research Notes in Mathematics 44, Pitman, ISBN 978-0-273-08446-4 • Facchini, Alberto (1998), Endomorphism rings and direct sum decompositions in some classes of modules, Birkhäuser Verlag, ISBN 3-7643-5908-0 • Faith, Carl (1976), Algebra. II. Ring theory., Grundlehren der Mathematischen Wissenschaften, No. 191. Springer-Verlag • Faith, Carl (1999), Rings and things and a fine array of twentieth century associative algebra, Mathematical Surveys and Monographs, 65. American Mathematical Society, ISBN 0-8218-0993-8 • Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V. (2004), Algebras, rings and modules. Vol. 1., Kluwer Academic Publishers, ISBN 1-4020-2690-0 • Puninski, Gennadi (2001), Serial rings, Kluwer Academic Publishers, ISBN 0-7923-7187-9

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253.7 Primary Sources • Eisenbud, David; Griffith, Phillip (1971), “The structure of serial rings.”, Pacific J. Math. 36: 109–121, doi:10.2140/pjm.1971.36.109 • Facchini, Alberto (1996), “Krull-Schmidt fails for serial modules”, Trans. Amer. Math. Soc. 348: 4561–4575 • Köthe, Gottfried (1935), “Verallgemeinerte Abelsche Gruppen mit hyperkomplexem Operatorenring. (German)", Math. Z. 39: 31–44, doi:10.1007/bf01201343 • Nakayama, Tadasi (1941), “On Frobeniusean algebras. II.”, Annals of Mathematics, Second Series 42: 1–21, doi:10.2307/1968984 • Příhoda, Pavel (2004), “Weak Krull-Schmidt theorem and direct sum decompositions of serial modules of finite Goldie dimension”, J. Algebra 281: 332–341, doi:10.1016/j.jalgebra.2004.06.027 • Příhoda, Pavel (2006), “A version of the weak Krull-Schmidt theorem for infinite direct sums of uniserial modules”, Comm. Algebra 34 (4): 1479–1487, doi:10.1080/00927870500455049 • Puninski, G. T. (2002), “Artinian and Noetherian serial rings.”, J. Math. Sci. (New York) 110: 2330–2347 • Puninski, Gennadi (2001), “Some model theory over a nearly simple uniserial domain and decompositions of serial modules”, J. Pure Appl. Algebra 163: 319–337, doi:10.1016/s0022-4049(00)00140-7 • Puninski, Gennadi (2001), “Some model theory over an exceptional uniserial ring and decompositions of serial modules”, Journal of the London Mathematical Society 64: 311–326, doi:10.1112/s0024610701002344 • Warfield, Robert B., Jr. (1975), “Serial rings and finitely presented modules.”, J. Algebra 37: 187–222., doi:10.1016/0021-8693(75)90074-5

Chapter 254

Serre’s criterion on normality In algebra, Serre’s criterion on normality, introduced by Jean-Pierre Serre, gives necessary and sufficient conditions for a commutative Noetherian ring A to be a normal ring. The criterion involves the following two conditions for A: • Rk : Ap is a regular local ring for any prime ideal p of height ≤ k. • Sk : depth Ap ≥ inf{k, ht(p)} for any prime ideal p .[1] The statement is: • A is a reduced ring ⇔ R0 , S1 hold. • A is a normal ring ⇔ R1 , S2 hold. • A is a Cohen–Macaulay ring ⇔ Sk hold for all k. Items 1, 3 trivially follow from the definitions. Item 2 is much deeper. For an integral domain, the criterion is due to Krull. The general case is due to Serre.

254.1 Proof 254.1.1

Sufficiency

(After EGA IV. Theorem 5.8.6.) Suppose A satisfies S 2 and R1 . Then A in particular satisfies S 1 and R0 ; hence, it is reduced. If pi , 1 ≤ i ≤ r are the minimal prime ideals of A, then the total ring of fractions K of A is the direct product of the residue fields κ(pi ) = Q(A/pi ) : see total ring of fractions of a reduced ring. That means we can write 1 = e1 + · · · + er where ei are idempotents in κ(pi ) and such that ei ej = 0, i ̸= j . Now, if A is integrally closed in K, then each ei is integral over A and so is in A; consequently, A is a direct product of integrally closed domains Aei's and we are done. Thus, it is enough to show that A is integrally closed in K. For this end, suppose

(f /g)n + a1 (f /g)n−1 + · · · + an = 0 where all f, g, ai's are in A and g is moreover a non-zerodivisor. We want to show:

f ∈ gA 773

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Now, the condition S 2 says that gA is unmixed of height one; i.e., each associated primes p of A/gA has height one. By the condition R1 , the localization Ap is integrally closed and so ϕ(f ) ∈ ϕ(g)Ap , where ϕ : A → Ap is the localization map, since the integral equation persists after localization. If gA = ∩i qi is the primary decomposition, then, for any i, the radical of qi is an associated prime p of A/gA and so f ∈ ϕ−1 (qi Ap ) = qi ; the equality here is because qi is a p -primary ideal. Hence, the assertion holds.

254.1.2

Necessity

Suppose A is a normal ring. For S 2 , let p be an associated prime of A/f A for a non-zerodivisor f; we need to show it has height one. Replacing A by a localization, we can assume A is a local ring with maximal ideal p . By definition, there is an element g in A such that p = {x ∈ A|xg ≡ 0 mod f A} and g ̸∈ f A . Put y = g/f in the total ring of fractions. If yp ⊂ p , then p is a faithful A[y] -module and is a finitely generated A-module; consequently, y is integral over A and thus in A, a contradiction. Hence, yp = A or p = f /gA , which implies p has height one (Krull’s principal ideal theorem). For R1 , we argue in the same way: let p be a prime ideal of height one. Localizing at p we assume p is a maximal ideal and the similar argument as above shows that p is in fact principal. Thus, A is a regular local ring. □

254.2 Notes [1] EGA IV, § 5.7.

254.3 References • Grothendieck, Alexandre; Dieudonné, Jean (1965). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Seconde partie”. Publications Mathématiques de l'IHÉS 24. doi:10.1007/bf02684322. MR 0199181. • H. Matsumura, Commutative algebra, 1970.

Chapter 255

Severi–Brauer variety In mathematics, a Severi–Brauer variety over a field K is an algebraic variety V which becomes isomorphic to a projective space over an algebraic closure of K. The varieties are associated to central simple algebras in such a way that the algebra splits over K if and only if the variety has a point rational over K.[1] Francesco Severi (1932) studied these varieties, and they are also named after Richard Brauer because of their close relation to the Brauer group. In dimension one, the Severi–Brauer varieties are conics. The corresponding central simple algebras are the quaternion algebras. The algebra (a,b)K corresponds to the conic C(a,b) with equation

z 2 = ax2 + by 2 and the algebra (a,b)K splits, that is, (a,b)K is isomorphic to a matrix algebra over K, if and only if C(a,b) has a point defined over K: this is in turn equivalent to C(a,b) being isomorphic to the projective line over K.[1][2] Such varieties are of interest not only in diophantine geometry, but also in Galois cohomology. They represent (at least if K is a perfect field) Galois cohomology classes in H 1 (PGLn) in the projective linear group, where n is the dimension of V. There is a short exact sequence 1 → GL1 → GLn → PGLn → 1 of algebraic groups. This implies a connecting homomorphism H 1 (PGLn) → H 2 (GL1 ) at the level of cohomology. Here H 2 (GL1) is identified with the Brauer group of K, while the kernel is trivial because H 1 (GLn) = {1} by an extension of Hilbert’s Theorem 90.[3][4] Therefore the Severi–Brauer varieties can be faithfully represented by Brauer group elements, i.e. classes of central simple algebras. Lichtenbaum showed that if X is a Severi–Brauer variety over K then there is an exact sequence

δ

0 → Pic(X) → Z→Br(K) → Br(K)(X) → 0 . Here the map δ sends 1 to the Brauer class corresponding to X.[2] As a consequence, we see that if the class of X has order d in the Brauer group then there is a divisor class of degree d on X. The associated linear system defines the d-dimensional embedding of X over a splitting field L.[5] 775

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255.1 References [1] Jacobson (1996) p.113 [2] Gille & Szamuely (2006) p.129 [3] Gille & Szamuely (2006) p.26 [4] Berhuy, Grégory (2010), An Introduction to Galois Cohomology and its Applications, London Mathematical Society Lecture Note Series 377, Cambridge University Press, p. 113, ISBN 0-521-73866-0, Zbl 1207.12003 [5] Gille & Szamuely (2006) p.131

• Artin, Michael (1982), “Brauer-Severi varieties”, Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981), Lecture Notes in Math. 917, Notes by A. Verschoren, Berlin, New York: Springer-Verlag, pp. 194– 210, doi:10.1007/BFb0092235, ISBN 978-3-540-11216-7, MR 657430, Zbl 0536.14006 • Hazewinkel, Michiel, ed. (2001), “Brauer–Severi variety”, Encyclopedia of Mathematics, Springer, ISBN 9781-55608-010-4 • Gille, Philippe; Szamuely, Tamás (2006), “Severi–Brauer varieties”, Central Simple Algebras and Galois Cohomology, Cambridge Studies in Advanced Mathematics 101, Cambridge University Press, pp. 114–134, ISBN 0-521-86103-9, MR 2266528, Zbl 1137.12001 • Jacobson, Nathan (1996), Finite-dimensional division algebras over fields, Berlin: Springer-Verlag, ISBN 3540-57029-2, Zbl 0874.16002 • Saltman, David J. (1999), Lectures on division algebras, Regional Conference Series in Mathematics 94, Providence, RI: American Mathematical Society, ISBN 0-8218-0979-2, Zbl 0934.16013 • Severi, Francesco (1932), “Un nuovo campo di ricerche nella geometria sopra una superficie e sopra una varietà algebrica”, Memorie della Reale Accademia d'Italia (in Italian) 3 (5), Reprinted in volume 3 of his collected works

255.2 Further reading • Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998), The book of involutions, Colloquium Publications 44, With a preface by J. Tits, Providence, RI: American Mathematical Society, ISBN 0-8218-0904-0, MR MR1632779, Zbl 0955.16001

255.3 External links • Expository paper on Galois descent (PDF)

Chapter 256

Simple algebra In mathematics, specifically in ring theory, an algebra is simple if it contains no non-trivial two-sided ideals and the multiplication operation is not uniformly zero (that is, there is some a and some b such that ab≠0). The second condition in the definition precludes the following situation; consider the algebra with the usual matrix operations: {[

] } 0 α α ∈ C 0 0

This is a one-dimensional algebra in which the product of any two elements is zero. This condition ensures that the algebra has a minimal nonzero left ideal, which simplifies certain arguments. An immediate example of simple algebras are division algebras, where every element has a multiplicative inverse, for instance, the real algebra of quaternions. Also, one can show that the algebra of n × n matrices with entries in a division ring is simple. In fact, this characterizes all finite-dimensional simple algebras up to isomorphism, i.e. any finite-dimensional simple algebra is isomorphic to a matrix algebra over some division ring. This result was given in 1907 by Joseph Wedderburn in his doctoral thesis, On hypercomplex numbers, which appeared in the Proceedings of the London Mathematical Society. Wedderburn’s thesis classified simple and semisimple algebras. Simple algebras are building blocks of semi-simple algebras: any finite-dimensional semi-simple algebra is a Cartesian product, in the sense of algebras, of simple algebras. Wedderburn’s result was later generalized to semisimple rings in the Artin–Wedderburn theorem.

256.1 Examples • A central simple algebra (sometimes called Brauer algebra) is a simple finite-dimensional algebra over a field F whose center is F.

256.2 Simple universal algebras In universal algebra, an abstract algebra A is called “simple” if and only if it has no nontrivial congruence relations, or equivalently, if every homomorphism with domain A is either injective or constant. As congruences on rings are characterized by their ideals, this notion is a straightforward generalization of the notion from ring theory: a ring is simple in the sense that it has no nontrivial ideals if and only if it is simple in the sense of universal algebra.

256.3 See also • simple group 777

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• simple ring • central simple algebra

256.4 References • A. A. Albert, Structure of algebras, Colloquium publications 24, American Mathematical Society, 2003, ISBN 0-8218-1024-3. P.37.

Chapter 257

Simple module In mathematics, specifically in ring theory, the simple modules over a ring R are the (left or right) modules over R that have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M. Simple modules form building blocks for the modules of finite length, and they are analogous to the simple groups in group theory. In this article, all modules will be assumed to be right unital modules over a ring R.

257.1 Examples Z-modules are the same as abelian groups, so a simple Z-module is an abelian group which has no non-zero proper subgroups. These are the cyclic groups of prime order. If I is a right ideal of R, then I is simple as a right module if and only if I is a minimal non-zero right ideal: If M is a non-zero proper submodule of I, then it is also a right ideal, so I is not minimal. Conversely, if I is not minimal, then there is a non-zero right ideal J properly contained in I. J is a right submodule of I, so I is not simple. If I is a right ideal of R, then R/I is simple if and only if I is a maximal right ideal: If M is a non-zero proper submodule of R/I, then the preimage of M under the quotient map R → R/I is a right ideal which is not equal to R and which properly contains I. Therefore I is not maximal. Conversely, if I is not maximal, then there is a right ideal J properly containing I. The quotient map R/I → R/J has a non-zero kernel which is not equal to R/I, and therefore R/I is not simple. Every simple R-module is isomorphic to a quotient R/m where m is a maximal right ideal of R.[1] By the above paragraph, any quotient R/m is a simple module. Conversely, suppose that M is a simple R-module. Then, for any non-zero element x of M, the cyclic submodule xR must equal M. Fix such an x. The statement that xR = M is equivalent to the surjectivity of the homomorphism R → M that sends r to xr. The kernel of this homomorphism is a right ideal I of R, and a standard theorem states that M is isomorphic to R/I. By the above paragraph, we find that I is a maximal right ideal. Therefore M is isomorphic to a quotient of R by a maximal right ideal. If k is a field and G is a group, then a group representation of G is a left module over the group ring k[G]. The simple k[G] modules are also known as irreducible representations. A major aim of representation theory is to understand the irreducible representations of groups.

257.2 Basic properties of simple modules The simple modules are precisely the modules of length 1; this is a reformulation of the definition. Every simple module is indecomposable, but the converse is in general not true. Every simple module is cyclic, that is it is generated by one element. Not every module has a simple submodule; consider for instance the Z-module Z in light of the first example above. Let M and N be (left or right) modules over the same ring, and let f : M → N be a module homomorphism. If M 779

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is simple, then f is either the zero homomorphism or injective because the kernel of f is a submodule of M. If N is simple, then f is either the zero homomorphism or surjective because the image of f is a submodule of N. If M = N, then f is an endomorphism of M, and if M is simple, then the prior two statements imply that f is either the zero homomorphism or an isomorphism. Consequently the endomorphism ring of any simple module is a division ring. This result is known as Schur’s lemma. The converse of Schur’s lemma is not true in general. For example, the Z-module Q is not simple, but its endomorphism ring is isomorphic to the field Q.

257.3 Simple modules and composition series Main article: Composition series If M is a module which has a non-zero proper submodule N, then there is a short exact sequence

0 → N → M → M /N → 0. A common approach to proving a fact about M is to show that the fact is true for the center term of a short exact sequence when it is true for the left and right terms, then to prove the fact for N and M/N. If N has a non-zero proper submodule, then this process can be repeated. This produces a chain of submodules

· · · ⊂ M2 ⊂ M1 ⊂ M. In order to prove the fact this way, one needs conditions on this sequence and on the modules Mi/Mi ₊ ₁. One particularly useful condition is that the length of the sequence is finite and each quotient module Mi/Mi ₊ ₁ is simple. In this case the sequence is called a composition series for M. In order to prove a statement inductively using composition series, the statement is first proved for simple modules, which form the base case of the induction, and then the statement is proved to remain true under an extension of a module by a simple module. For example, the Fitting lemma shows that the endomorphism ring of a finite length indecomposable module is a local ring, so that the strong Krull-Schmidt theorem holds and the category of finite length modules is a Krull-Schmidt category. The Jordan–Hölder theorem and the Schreier refinement theorem describe the relationships amongst all composition series of a single module. The Grothendieck group ignores the order in a composition series and views every finite length module as a formal sum of simple modules. Over semisimple rings, this is no loss as every module is a semisimple module and so a direct sum of simple modules. Ordinary character theory provides better arithmetic control, and uses simple CG modules to understand the structure of finite groups G. Modular representation theory uses Brauer characters to view modules as formal sums of simple modules, but is also interested in how those simple modules are joined together within composition series. This is formalized by studying the Ext functor and describing the module category in various ways including quivers (whose nodes are the simple modules and whose edges are composition series of non-semisimple modules of length 2) and Auslander–Reiten theory where the associated graph has a vertex for every indecomposable module.

257.4 The Jacobson density theorem Main article: Jacobson density theorem An important advance in the theory of simple modules was the Jacobson density theorem. The Jacobson density theorem states: Let U be a simple right R-module and write D = EndR(U). Let A be any D-linear operator on U and let X be a finite D-linearly independent subset of U. Then there exists an element r of R such that x·A = x·r for all x in X.[2]

257.5. SEE ALSO

781

In particular, any primitive ring may be viewed as (that is, isomorphic to) a ring of D-linear operators on some D-space. A consequence of the Jacobson density theorem is Wedderburn’s theorem; namely that any right artinian simple ring is isomorphic to a full matrix ring of n by n matrices over a division ring for some n. This can also be established as a corollary of the Artin–Wedderburn theorem.

257.5 See also • Semisimple modules are modules that can be written as a sum of simple submodules • Irreducible ideal • Irreducible representation

257.6 References [1] Herstein, Non-commutative Ring Theory, Lemma 1.1.3 [2] Isaacs, Theorem 13.14, p. 185

Chapter 258

Simple ring In abstract algebra, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra. Rings which are simple as rings but not as modules do exist: the full matrix ring over a field does not have any nontrivial ideals (since any ideal of M(n,R) is of the form M(n,I) with I an ideal of R), but has nontrivial left ideals (namely, the sets of matrices which have some fixed zero columns). According to the Artin–Wedderburn theorem, every simple ring that is left or right Artinian is a matrix ring over a division ring. In particular, the only simple rings that are a finite-dimensional vector space over the real numbers are rings of matrices over either the real numbers, the complex numbers, or the quaternions. Any quotient of a ring by a maximal ideal is a simple ring. In particular, a field is a simple ring. A ring R is simple if and only its opposite ring Ro is simple. An example of a simple ring that is not a matrix ring over a division ring is the Weyl algebra.

258.1 Wedderburn’s theorem Wedderburn’s theorem characterizes simple rings with a unit and a minimal left ideal. (The left Artinian condition is a generalization of the second assumption.) Namely it says that every such ring is, up to isomorphism, a ring of n × n matrices over a division ring. Let D be a division ring and M(n,D) be the ring of matrices with entries in D. It is not hard to show that every left ideal in M(n,D) takes the following form: {M ∈ M(n,D) | The n1 ...nk-th columns of M have zero entries}, for some fixed {n1 ,...,nk} ⊂ {1, ..., n}. So a minimal ideal in M(n,D) is of the form {M ∈ M(n,D) | All but the k-th columns have zero entries}, for a given k. In other words, if I is a minimal left ideal, then I = (M(n,D)) e where e is the idempotent matrix with 1 in the (k, k) entry and zero elsewhere. Also, D is isomorphic to e(M(n,D))e. The left ideal I can be viewed as a right-module over e(M(n,D))e, and the ring M(n,D) is clearly isomorphic to the algebra of homomorphisms on this module. The above example suggests the following lemma: Lemma. A is a ring with identity 1 and an idempotent element e where AeA = A. Let I be the left ideal Ae, considered as a right module over eAe. Then A is isomorphic to the algebra of homomorphisms on I, denoted by Hom(I). Proof: We define the “left regular representation” Φ : A → Hom(I) by Φ(a)m = am for m ∈ I. Φ is injective because if a · I = aAe = 0, then aA = aAeA = 0, which implies a = a · 1 = 0. For surjectivity, let T ∈ Hom(I). Since AeA = A, the unit 1 can be expressed as 1 = ∑aiebi. So 782

258.2. SEE ALSO

783

T(m) = T(1·m) = T(∑aiebim) = ∑ T(aieebim) = ∑ T(aie) ebim = [ ∑T(aie)ebi]m. Since the expression [∑T(aie)ebi] does not depend on m, Φ is surjective. This proves the lemma. Wedderburn’s theorem follows readily from the lemma. Theorem (Wedderburn). If A is a simple ring with unit 1 and a minimal left ideal I, then A is isomorphic to the ring of n × n matrices over a division ring. One simply has to verify the assumptions of the lemma hold, i.e. find an idempotent e such that I = Ae, and then show that eAe is a division ring. The assumption A = AeA follows from A being simple.

258.2 See also • simple (algebra)

258.3 References • D.W. Henderson, A short proof of Wedderburn’s theorem, Amer. Math. Monthly 72 (1965), 385-386. • Lam, Tsit-Yuen (2001), A First Course in Noncommutative Rings (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-95325-0, MR 1838439

Chapter 259

Simplicial commutative ring In algebra, a simplicial commutative ring is a commutative monoid in the category of simplicial abelian groups, or, equivalently, a simplicial object in the category of commutative rings. If A is a simplicial commutative ring, then it can be shown that π0 A is a commutative ring and πi A are modules over that ring (in fact, π∗ A is a graded ring over π0 A .) A topology-counterpart of this notion is a commutative ring spectrum.

259.1 Graded ring structure Let A be a simplicial commutative ring. Then the ring structure of A gives π∗ A = ⊕i≥0 πi A the structure of a graded-commutative graded ring as follows. By the Dold–Kan correspondence, π∗ A is the homology of the chain complex corresponding to A; in particular, it is a graded abelian group. Next, to multiply two elements, writing S 1 for the circle, let x : (S 1 )∧i → A, y : (S 1 )∧j → A be two maps. Then the composition

(S 1 )∧i × (S 1 )∧j → A × A → A the second map the multiplication of A, induces (S 1 )∧i ∧ (S 1 )∧j → A . This in turn gives an element in πi+j A . We have thus defined the graded multiplication πi A × πj A → πi+j A . It is associative since the smash product is. It is graded-commutative (i.e., xy = (−1)|x||y| yx ) since the involution S 1 ∧ S 1 → S 1 ∧ S 1 introduces minus sign.

259.2 Spec By definition, the category of affine derived schemes is the opposite category of the category of simplicial commutative rings; an object corresponding to A will be denoted by Spec A .

259.3 See also • En-ring

259.4 References

• http://mathoverflow.net/questions/118500/what-is-a-simplicial-commutative-ring-from-the-point-of-view-of-homotopy-theo

• http://mathoverflow.net/questions/45273/what-facts-in-commutative-algebra-fail-miserably-for-simplicial-commutative-ring 784

259.4. REFERENCES • A. Mathew, Simplicial commutative rings, I. • B. Toën, Simplicial presheaves and derived algebraic geometry • P. Goerss and K. Schemmerhorn, Model categories and simplicial methods

785

Chapter 260

Singular submodule In the branches of abstract algebra known as ring theory and module theory, each right (resp. left) R module M has a singular submodule consisting of elements whose annihilators are essential right (resp. left) ideals in R. In set notation it is usually denoted as Z(M ) = {m ∈ M | ann(m) ⊆e R} . For general rings, Z(M ) is a good generalization of the torsion submodule tors(M) which is most often defined for domains. In the case that R is a commutative domain, tors(M ) = Z(M ) . If R is any ring, Z(RR ) is defined considering R as a right module, and in this case Z(RR ) is a twosided ideal of R called the right singular ideal of R. Similarly the left handed analogue Z(R R) is defined. It is possible for Z(RR ) ̸= Z(R R) .

260.1 Definitions Here are several definitions used when studying singular submodule and singular ideals. In the following, M is an R module: • M is called a singular module if Z(M ) = M . • M is called a nonsingular module if Z(M ) = {0} . • R is called right nonsingular if Z(RR ) = {0} . A left nonsingular ring is defined similarly, using the left singular ideal, and it is entirely possible for a ring to be right-but-not-left nonsingular. In rings with unity it is always the case that Z(RR ) ⊊ R , and so “right singular ring” is not usually defined the same way as singular modules are. Some authors have used “singular ring” to mean “has a nonzero singular ideal”, however this usage is not consistent with the usage of the adjectives for modules.

260.2 Properties Some general properties of the singular submodule include: • Z(M ) · soc(M ) = {0} where soc(M ) denotes the socle of M. • If f is a homomorphism of R modules from M to N, then f (Z(M )) ⊆ Z(N ) . • If N is a submodule of M, then Z(N ) = N ∩ Z(M ) . • The properties “singular” and “nonsingular” are Morita invariant properties. • The singular ideals of a ring contain central nilpotent elements of the ring. Consequently the singular ideal of a commutative ring contains the nilradical of the ring. 786

260.3. EXAMPLES

787

• A general property of the torsion submodule is that t(M /t(M )) = {0} , but this does not necessarily hold for the singular submodule. However if R is a right nonsingular ring, then Z(M /Z(M )) = {0} . • If N is an essential submodule of M (both right modules) then M/N is singular. If M is a free module, or if R is right nonsingular, then the converse is true. • A semisimple module is nonsingular if and only if it is a projective module. • If R is a right self-injective ring, then Z(RR ) = J(R) , where J(R) is the Jacobson radical of R.

260.3 Examples Right nonsingular rings are a very broad class, including reduced rings, right (semi)hereditary rings, von Neumann regular rings, domains, semisimple rings, Baer rings and right Rickart rings. For commutative rings, being nonsingular is equivalent to being a reduced ring.

260.4 Important theorems Johnson’s Theorem (due to R. E. Johnson (Lam 1999, p. 376)) contains several important equivalences. For any ring R, the following are equivalent: 1. R is right nonsingular. 2. The injective hull E(RR) is a nonsingular right R module. 3. The endomorphism ring S = End(E(RR )) is a semiprimitive ring (that is, J(S) = {0} ). 4. The maximal right ring of quotients Qrmax (R) is von Neumann regular. Right nonsingularity has a strong interaction with right self injective rings as well. Theorem: If R is a right self injective ring, then the following conditions on R are equivalent: right nonsingular, von Neumann regular, right semihereditary, right Rickart, Baer, semiprimitive. (Lam 1999, p. 262) The paper (Zelmanowitz 1983) used nonsingular modules to characterize the class of rings whose maximal right ring of quotients have a certain structure. Theorem: If R is a ring, then Qrmax (R) is a right full linear ring if and only if R has a nonsingular, faithful, uniform module. Moreover, Qrmax (R) is a finite direct product of full linear rings if and only if R has a nonsingular, faithful module with finite uniform dimension.

260.5 Textbooks • Goodearl, K. R. (1976), Ring theory: Nonsingular rings and modules, Pure and Applied Mathematics, No. 33, New York: Marcel Dekker Inc., pp. viii+206, MR 0429962 • Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294

260.6 Primary sources • Zelmanowitz, J. M. (1983), “The structure of rings with faithful nonsingular modules”, Trans. Amer. Math. Soc. 278 (1): 347–359, doi:10.2307/1999320, ISSN 0002-9947, MR 697079 84d:16030)

Chapter 261

Skolem–Noether theorem In ring theory, a branch of mathematics, the Skolem–Noether theorem characterizes the automorphisms of simple rings. It is a fundamental result in the theory of central simple algebras. The theorem was first published by Thoralf Skolem in 1927 in his paper Zur Theorie der assoziativen Zahlensysteme (German: On the theory of associative number systems) and later rediscovered by Emmy Noether.

261.1 Statement In a general formulation, let A and B be simple rings, and let k be the centre of B. Notice that k is a field since given x nonzero in k, the simplicity of B implies that the nonzero two-sided ideal Bx is the whole of B, and hence that x is a unit. Suppose further that the dimension of B over k is finite, i.e. that B is a central simple algebra. Then given k-algebra homomorphisms f, g : A → B there exists a unit b in B such that for all a in A[1][2] g(a) = b · f(a) · b−1 . In particular, every automorphism of a central simple k-algebra is an inner automorphism.[3][4]

261.2 Proof First suppose B = Mn (k) = Endk (k n ) . Then f and g define the actions of A on k n ; let Vf , Vg denote the A-modules thus obtained. Any two simple A-modules are isomorphic and Vf , Vg are finite direct sums of simple A-modules. Since they have the same dimension, it follows that there is an isomorphism of A-modules b : Vg → Vf . But such b must be an element of Mn (k) = B . For the general case, note that B ⊗ B op is a matrix algebra and thus by the first part this algebra has an element b such that

(f ⊗ 1)(a ⊗ z) = b(g ⊗ 1)(a ⊗ z)b−1 for all a ∈ A and z ∈ B op . Taking a = 1 , we find 1 ⊗ z = b(1 ⊗ z)b−1 for all z. That is to say, b is in ZB⊗B op (k ⊗ B op ) = B ⊗ k and so we can write b = b′ ⊗ 1 . Taking z = 1 this time we find 788

261.3. NOTES

789

f (a) = b′ g(a)b′−1 which is what was sought.

261.3 Notes [1] Lorenz (2008) p.173 [2] Farb, Benson; Dennis, R. Keith (1993). Noncommutative Algebra. Springer. ISBN 9780387940571. [3] Gille & Szamuely (2006) p.40 [4] Lorenz (2008) p.174

261.4 References • Skolem, Thoralf (1927). “Zur Theorie der assoziativen Zahlensysteme”. Skrifter Oslo (in German) (12): 50. JFM 54.0154.02. • A discussion in Chapter IV of Milne, class field theory • Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001. • Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. ISBN 978-0-387-72487-4. Zbl 1130.12001.

Chapter 262

Socle (mathematics) In mathematics, the term socle has several related meanings.

262.1 Socle of a group In the context of group theory, the socle of a group G, denoted soc(G), is the subgroup generated by the minimal normal subgroups of G. It can happen that a group has no minimal non-trivial normal subgroup (that is, every nontrivial normal subgroup properly contains another such subgroup) and in that case the socle is defined to be the subgroup generated by the identity. The socle is a direct product of minimal normal subgroups.[1] As an example, consider the cyclic group Z12 with generator u, which has two minimal normal subgroups, one generated by u 4 (which gives a normal subgroup with 3 elements) and the other by u 6 (which gives a normal subgroup with 2 elements). Thus the socle of Z12 is the group generated by u 4 and u 6 , which is just the group generated by u 2. The socle is a characteristic subgroup, and hence a normal subgroup. It is not necessarily transitively normal, however. If a group G is a finite solvable group, then the socle can be expressed as a product of elementary abelian p-groups. Thus, in this case, it is just a product of copies of Z/pZ for various p where the same p may occur multiple times in the product.

262.2 Socle of a module In the context of module theory and ring theory the socle of a module M over a ring R is defined to be the sum of the minimal nonzero submodules of M. It can be considered as a dual notion to that of the radical of a module. In set notation,

soc(M ) =

∑

{N | N of submodule simple a is M }.

Equivalently,

soc(M ) =

∩

{E | E of submodule essential an is M }.

The socle of a ring R can refer to one of two sets in the ring. Considering R as a right R module, soc(RR) is defined, and considering R as a left R module, soc(RR) is defined. Both of these socles are ring ideals, and it is known they are not necessarily equal. • If M is an Artinian module, soc(M) is itself an essential submodule of M. • A module is semisimple if and only if soc(M) = M. Rings for which soc(M) = M for all M are precisely semisimple rings. 790

262.3. SOCLE OF A LIE ALGEBRA

791

• M is a finitely cogenerated module if and only if soc(M) is finitely generated and soc(M) is an essential submodule of M. • Since the sum of semisimple modules is semisimple, the socle of a module could also be defined as the unique maximal semi-simple submodule. • From the definition of rad(R), it is easy to see that rad(R) annihilates soc(R). If R is a finite-dimensional unital algebra and M a finitely generated R-module then the socle consists precisely of the elements annihilated by the Jacobson radical of R.[2]

262.3 Socle of a Lie algebra In the context of Lie algebras, a socle of a symmetric Lie algebra is the eigenspace of its structural automorphism which corresponds to the eigenvalue −1. (A symmetric Lie algebra decomposes into the direct sum of its socle and cosocle.)[3]

262.4 See also • Injective hull • Radical of a module • Cosocle

262.5 References [1] Robinson 1996, p.87. [2] J. L. Alperin; Rowen B. Bell, Groups and Representations, 1995, ISBN 0-387-94526-1, p. 136 [3] Mikhail Postnikov, Geometry VI: Riemannian Geometry, 2001, ISBN 3540411089,p. 98

• Alperin, J.L.; Bell, Rowen B. (1995). Groups and Representations. Springer-Verlag. p. 136. ISBN 0-38794526-1. • Anderson, Frank Wylie; Fuller, Kent R. (1992). Rings and Categories of Modules. Springer-Verlag. ISBN 978-0-387-97845-1. • Robinson, Derek J. S. (1996), A course in the theory of groups, Graduate Texts in Mathematics 80 (2 ed.), New York: Springer-Verlag, pp. xviii+499, ISBN 0-387-94461-3, MR 1357169

Chapter 263

Spacetime algebra In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cℓ₁,₃(R), or equivalently the geometric algebra G4 = G(M4), which can be particularly closely associated with the geometry of special relativity and relativistic spacetime. It is a vector space allowing not just vectors, but also bivectors (directed quantities associated with particular planes, such as areas, or rotations) or multivectors (quantities associated with particular hyper-volumes) to be combined, as well as rotated, reflected, or Lorentz boosted. It is also the natural parent algebra of spinors in special relativity. These properties allow many of the most important equations in physics to be expressed in particularly simple forms, and can be very helpful towards a more geometric understanding of their meanings.

263.1 Structure The spacetime algebra is built up from combinations of one time-like basis vector γ0 and three orthogonal space-like vectors, {γ1 , γ2 , γ3 } , under the multiplication rule γµ γν + γν γµ = 2ηµν where ηµν is the Minkowski metric with signature (+ − − −) Thus γ02 = +1 , γ12 = γ22 = γ32 = −1 , otherwise γµ γν = −γν γµ . The basis vectors γk share these properties with the Dirac matrices, but no explicit matrix representation is utilized in STA. This generates a basis of one scalar {1} , four vectors {γ0 , γ1 , γ2 , γ3 } , six bivectors {γ0 γ1 , γ0 γ2 , γ0 γ3 , γ1 γ2 , γ2 γ3 , γ3 γ1 } , four pseudovectors {iγ0 , iγ1 , iγ2 , iγ3 } and one pseudoscalar {i} , where i = γ0 γ1 γ2 γ3 .

263.2 Reciprocal frame Associated with the orthogonal basis {γµ } is the reciprocal basis {γ µ =

1 γµ }

for all µ =0,...,3, satisfying the relation

γµ · γ ν = δµ ν These reciprocal frame vectors differ only by a sign, with γ 0 = γ0 , and γ k = −γk for k =1,...,3. A vector may be represented in either upper or lower index coordinates a = aµ γµ = aµ γ µ with summation over µ =0,...,3, according to the Einstein notation, where the coordinates may be extracted by taking dot products with the basis vectors or their reciprocals. a · γ ν = aν a · γν = aν 792

263.3. SPACETIME GRADIENT

793

263.3 Spacetime gradient The spacetime gradient, like the gradient in a Euclidean space, is defined such that the directional derivative relationship is satisfied:

a · ∇F (x) = lim

τ →0

F (x + aτ ) − F (x) τ

This requires the definition of the gradient to be

∇ = γµ

∂ = γ µ ∂µ . ∂xµ

Written out explicitly with x = ctγ0 + xk γk , these partials are

∂0 =

1 ∂ , c ∂t

∂k =

∂ ∂xk

263.4 Spacetime split In spacetime algebra, a spacetime split is a projection from 4D space into (3+1)D space with a chosen reference frame by means of the following two operations: • a collapse of the chosen time axis, yielding a 3D space spanned by bivectors, and • a projection of the 4D space onto the chosen time axis, yielding a 1D space of scalars.[3] This is achieved by pre or post multiplication by the timelike basis vector γ0 , which serves to split a four vector into a scalar timelike and a bivector spacelike component. With x = xµ γµ we have

xγ0 = x0 + xk γk γ0 γ0 x = x0 − xk γk γ0 As these bivectors γk γ0 square to unity, they serve as a spatial basis. Utilizing the Pauli matrix notation, these are written σk = γk γ0 . Spatial vectors in STA are denoted in boldface; then with x = xk σk the γ0 -spacetime split xγ0 and its reverse γ0 x are:

xγ0 = x0 + xk σk = x0 + x γ0 x = x0 − xk σk = x0 − x

263.5 Multivector division The spacetime algebra is not a division algebra, because it contains idempotent elements 12 (1 ± γ0 γi ) and zero divisors: (1 + γ0 γi )(1 − γ0 γi ) = 0 . These can be interpreted as projectors onto the light-cone and orthogonality relations for such projectors, respectively. But in general it is possible to divide one multivector quantity by another, and make sense of the result: so, for example, a directed area divided by a vector in the same plane gives another vector, orthogonal to the first.

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CHAPTER 263. SPACETIME ALGEBRA

263.6 Spacetime algebra description of non-relativistic physics 263.6.1

Non-relativistic quantum mechanics

Spacetime algebra allows to describe the Pauli particle in terms of a real theory in place of a matrix theory. The matrix theory description of the Pauli particle is:[4]

iℏ ∂t Ψ = HS Ψ −

eℏ σ ˆ · BΨ 2mc

where i is the imaginary unit with no geometric interpretation, σ ˆi are the Pauli matrices (with the ‘hat’ notation indicating that σ ˆ is a matrix operator and not an element in the geometric algebra), and HS is the Schrödinger Hamiltonian. In the spacetime algebra the Pauli particle is described by the real Pauli–Schrödinger equation:[4]

∂t ψ iσ3 ℏ = HS ψ −

eℏ Bψσ3 2mc

where now i is the unit pseudoscalar i = σ1 σ2 σ3 , and ψ and σ3 are elements of the geometric algebra, with ψ an even multi-vector; HS is again the Schrödinger Hamiltonian. Hestenes refers to this as the real Pauli–Schrödinger theory to emphasize that this theory reduces to the Schrödinger theory if the term that includes the magnetic field is dropped.

263.7 Spacetime algebra description of relativistic physics 263.7.1

Relativistic quantum mechanics

The relativistic quantum wavefunction is sometimes expressed as a spinor field, i.e.

1

ψ = e 2 (µ+βi+ϕ) where ϕ is a bivector, and[5][6]

1

ψ = R(ρeiβ ) 2 where according to its derivation by David Hestenes, ψ = ψ(x) is an even multivector-valued function on spacetime, R = R(x) is a unimodular spinor (or “rotor”[7] ), and ρ = ρ(x) and β = β(x) are scalar-valued functions.[5] This equation is interpreted as connecting spin with the imaginary pseudoscalar.[8] R is viewed as a Lorentz rotation ˜ ,[7] where the tilde symbol which a frame of vectors γµ into another frame of vectors eµ by the operation eµ = Rγµ R indicates the reverse (the reverse is often also denoted by the dagger symbol, see also Rotations in geometric algebra). This has been extended to provide a framework for locally varying vector- and scalar-valued observables and support for the Zitterbewegung interpretation of quantum mechanics originally proposed by Schrödinger. Hestenes has compared his expression for ψ with Feynman’s expression for it in the path integral formulation:

ψ = eiΦλ /ℏ where Φλ is the classical action along the λ -path.[5] Spacetime algebra allows to describe the Dirac particle in terms of a real theory in place of a matrix theory. The matrix theory description of the Dirac particle is:[9]

γˆ µ (j∂µ − eAµ )|ψ⟩ = m|ψ⟩

263.8. SEE ALSO

795

where γˆ are the Dirac matrices. In the spacetime algebra the Dirac particle is described by the equation:[9]

∇ψ iσ3 − Aψ = mψγ0 Here, ψ and σ3 are elements of the geometric algebra, and ∇ = γ µ ∂µ is the spacetime vector derivative.

263.7.2

A new formulation of General Relativity

Lasenby, Doran, and Gull of Cambridge University have proposed a new formulation of gravity, termed gauge theory gravity (GTG), wherein spacetime algebra is used to induce curvature on Minkowski space while admitting a gauge symmetry under “arbitrary smooth remapping of events onto spacetime” (Lasenby, et al.); a nontrivial proof then leads to the geodesic equation,

1 d R = (Ω − ω)R dτ 2 and the covariant derivative

1 D τ = ∂τ + ω 2 where ω is the connexion associated with the gravitational potential, and Ω is an external interaction such as an electromagnetic field. The theory shows some promise for the treatment of black holes, as its form of the Schwarzschild solution does not break down at singularities; most of the results of general relativity have been mathematically reproduced, and the relativistic formulation of classical electrodynamics has been extended to quantum mechanics and the Dirac equation.

263.8 See also • Geometric algebra • Dirac algebra • Dirac equation • General relativity

263.9 References • A. Lasenby, C. Doran, & S. Gull, “Gravity, gauge theories and geometric algebra,” Phil. Trans. R. Lond. A 356: 487–582 (1998). • Chris Doran and Anthony Lasenby (2003). Geometric Algebra for Physicists, Cambridge Univ. Press. ISBN 0-521-48022-1 • David Hestenes (1966). Space-Time Algebra, Gordon & Breach. • David Hestenes and Sobczyk, G. (1984). Clifford Algebra to Geometric Calculus, Springer Verlag ISBN 90277-1673-0 • David Hestenes (1973). “Local observables in the Dirac theory”, J. Math. Phys. Vol. 14, No. 7. • David Hestenes (1967). “Real Spinor Fields”, Journal of Mathematical Physics, 8 No. 4, (1967), 798–808.

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[1] A. N. Lasenby, C. J. L. Doran: Geometric algebra, Dirac wavefunctions and black holes. In: Peter Gabriel Bergmann, Venzo De Sabbata (eds.): Advances in the interplay between quantum and gravity physics, Springer, 2002, ISBN 978-14020-0593-0, pp. 256-283, p. 257 [2] A. N. Lasenby, C. J. L. Doran: Geometric algebra, Dirac wavefunctions and black holes. In: Peter Gabriel Bergmann, Venzo De Sabbata (eds.): Advances in the interplay between quantum and gravity physics, Springer, 2002, ISBN 978-14020-0593-0, pp. 256-283, p. 259 [3] John W. Arthur: Understanding Geometric Algebra for Electromagnetic Theory (IEEE Press Series on Electromagnetic Wave Theory), Wiley, 2011, ISBN 978-0-470-94163-8, p. 180 [4] See eqs. (75) and (81) in: D. Hestenes: Oersted Medal Lecture [5] See eq. (3.1) and similarly eq. (4.1), and subsequent pages, in: D. Hestenes: On decoupling probability from kinematics in quantum mechanics, In: P.F. Fougère (ed.): Maximum Entropy and Bayesian Methods, Kluwer Academic Publishers, 1990, pp. 161–183 (PDF) [6] See also eq. (5.13) of S. Gull, A. Lasenby, C. Doran: Imaginary numbers are not real – the geometric algebra of spacetime, 1993 [7] See eq. (205) in: D. Hestenes: Spacetime physics with geometric algebra, American Journal of Physics, vol. 71, no. 6, June 2003, pp. 691 ff., DOI 10.1119/1.1571836 (abstract, full text) [8] D. Hestenes, Oersted Medal Lecture 2002: Reforming the mathematical language of physics, DOI 10.1119/1.1522700 (abstract, full text) [9] See eqs. (3.43) and (3.44) in: Chris Doran, Anthony Lasenby, Stephen Gull, Shyamal Somaroo, Anthony Challinor: Spacetime algebra and electron physics, in: Peter W. Hawkes (ed.): Advances in Imaging and Electron Physics, Vol. 95, Academic Press, 1996, ISBN 0-12-014737-8, p. 272–386, p. 292

263.10 External links • Imaginary numbers are not real – the geometric algebra of spacetime, a tutorial introduction to the ideas of geometric algebra, by S. Gull, A. Lasenby, C. Doran • Physical Applications of Geometric Algebra course-notes, see especially part 2. • Cambridge University Geometric Algebra group • Geometric Calculus research and development

Chapter 264

Spectrum of a ring For the concept of ring spectrum in homotopy theory, see Ring spectrum. In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.

264.1 Zariski topology For any ideal I of R, define VI to be the set of prime ideals containing I. We can put a topology on Spec(R) by defining the collection of closed sets to be

{VI : I of ideal an is R}. This topology is called the Zariski topology. A basis for the Zariski topology can be constructed as follows. For f∈R, define Df to be the set of prime ideals of R not containing f. Then each Df is an open subset of Spec(R), and {Df : f ∈ R} is a basis for the Zariski topology. Spec(R) is a compact space, but almost never Hausdorff: in fact, the maximal ideals in R are precisely the closed points in this topology. However, Spec(R) is always a Kolmogorov space. It is also a spectral space.

264.2 Sheaves and schemes Given the space X=Spec(R) with the Zariski topology, the structure sheaf OX is defined on the Df by setting Γ(Df, OX) = Rf, the localization of R at the multiplicative system {1,f,f 2 ,f 3 ,...}. It can be shown that this satisfies the necessary axioms to be a B-Sheaf. Next, if U is the union of {Dfi}i∈I, we let Γ(U,OX) = limi∈I Rfi, and this produces a sheaf; see the Gluing axiom article for more detail. If R is an integral domain, with field of fractions K, then we can describe the ring Γ(U,OX) more concretely as follows. We say that an element f in K is regular at a point P in X if it can be represented as a fraction f = a/b with b not in P. Note that this agrees with the notion of a regular function in algebraic geometry. Using this definition, we can describe Γ(U,OX) as precisely the set of elements of K which are regular at every point P in U. If P is a point in Spec(R), that is, a prime ideal, then the stalk at P equals the localization of R at P, and this is a local ring. Consequently, Spec(R) is a locally ringed space. Every locally ringed space isomorphic to one of this form is called an affine scheme. General schemes are obtained by “gluing together” several affine schemes. 797

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CHAPTER 264. SPECTRUM OF A RING

264.3 Functoriality It is useful to use the language of category theory and observe that Spec is a functor. Every ring homomorphism f : R → S induces a continuous map Spec(f) : Spec(S) → Spec(R) (since the preimage of any prime ideal in S is a prime ideal in R). In this way, Spec can be seen as a contravariant functor from the category of commutative rings to the category of topological spaces. Moreover for every prime P the homomorphism f descends to homomorphisms Of −₁ ₍P₎ → OP of local rings. Thus Spec even defines a contravariant functor from the category of commutative rings to the category of locally ringed spaces. In fact it is the universal such functor and this can be used to define the functor Spec up to natural isomorphism. The functor Spec yields a contravariant equivalence between the category of commutative rings and the category of affine schemes; each of these categories is often thought of as the opposite category of the other.

264.4 Motivation from algebraic geometry Following on from the example, in algebraic geometry one studies algebraic sets, i.e. subsets of K n (where K is an algebraically closed field) that are defined as the common zeros of a set of polynomials in n variables. If A is such an algebraic set, one considers the commutative ring R of all polynomial functions A → K. The maximal ideals of R correspond to the points of A (because K is algebraically closed), and the prime ideals of R correspond to the subvarieties of A (an algebraic set is called irreducible or a variety if it cannot be written as the union of two proper algebraic subsets). The spectrum of R therefore consists of the points of A together with elements for all subvarieties of A. The points of A are closed in the spectrum, while the elements corresponding to subvarieties have a closure consisting of all their points and subvarieties. If one only considers the points of A, i.e. the maximal ideals in R, then the Zariski topology defined above coincides with the Zariski topology defined on algebraic sets (which has precisely the algebraic subsets as closed sets). One can thus view the topological space Spec(R) as an “enrichment” of the topological space A (with Zariski topology): for every subvariety of A, one additional non-closed point has been introduced, and this point “keeps track” of the corresponding subvariety. One thinks of this point as the generic point for the subvariety. Furthermore, the sheaf on Spec(R) and the sheaf of polynomial functions on A are essentially identical. By studying spectra of polynomial rings instead of algebraic sets with Zariski topology, one can generalize the concepts of algebraic geometry to non-algebraically closed fields and beyond, eventually arriving at the language of schemes.

264.5 Global Spec There is a relative version of the functor Spec called global Spec, or relative Spec, and denoted by Spec. For a scheme Y, and a quasi-coherent sheaf of OY-algebras A, there is a unique scheme SpecA, and a morphism f : Spec A → Y such that for every open affine U ⊆ Y , there is an isomorphism induced by f: f −1 (U ) ∼ = Spec A(U ) , and such that for open affines U ⊆ V , the inclusion f −1 (U ) → f −1 (V ) induces the restriction map A(V ) → A(U ). That is, as ring homomorphisms induce opposite maps of spectra, the restriction maps of a sheaf of algebras induce the inclusion maps of the spectra that make up the Spec of the sheaf.

264.6 Representation theory perspective From the perspective of representation theory, a prime ideal I corresponds to a module R/I, and the spectrum of a ring corresponds to irreducible cyclic representations of R, while more general subvarieties correspond to possibly reducible representations that need not be cyclic. Recall that abstractly, the representation theory of a group is the study of modules over its group algebra. The connection to representation theory is clearer if one considers the polynomial ring R = K[x1 , . . . , xn ] or, without a basis, R = K[V ]. As the latter formulation makes clear, a polynomial ring is the group algebra over a

264.7. FUNCTIONAL ANALYSIS PERSPECTIVE

799

vector space, and writing in terms of xi corresponds to choosing a basis for the vector space. Then an ideal I, or equivalently a module R/I, is a cyclic representation of R (cyclic meaning generated by 1 element as an R-module; this generalizes 1-dimensional representations). In the case that the field is algebraically closed (say, the complex numbers), every maximal ideal corresponds to a point in n-space, by the nullstellensatz (the maximal ideal generated by (x1 − a1 ), (x2 − a2 ), . . . , (xn − an ) corresponds to the point (a1 , . . . , an ) ). These representations of K[V ] are then parametrized by the dual space V ∗ , the covector being given by sending each xi to the corresponding ai . Thus a representation of K n (K-linear maps K n → K ) is given by a set of n numbers, or equivalently a covector K n → K. Thus, points in n-space, thought of as the max spec of R = K[x1 , . . . , xn ], correspond precisely to 1-dimensional representations of R, while finite sets of points correspond to finite-dimensional representations (which are reducible, corresponding geometrically to being a union, and algebraically to not being a prime ideal). The non-maximal ideals then correspond to infinite-dimensional representations.

264.7 Functional analysis perspective Main article: Spectrum (functional analysis) For more details on this topic, see Algebra representation § Weights. The term “spectrum” comes from the use in operator theory. Given a linear operator T on a finite-dimensional vector space V, one can consider the vector space with operator as a module over the polynomial ring in one variable R=K[T], as in the structure theorem for finitely generated modules over a principal ideal domain. Then the spectrum of K[T] (as a ring) equals the spectrum of T (as an operator). Further, the geometric structure of the spectrum of the ring (equivalently, the algebraic structure of the module) captures the behavior of the spectrum of the operator, such as algebraic multiplicity and geometric multiplicity. For instance, for the 2×2 identity matrix has corresponding module:

K[T ]/(T − 1) ⊕ K[T ]/(T − 1) the 2×2 zero matrix has module

K[T ]/(T − 0) ⊕ K[T ]/(T − 0), showing geometric multiplicity 2 for the zero eigenvalue, while a non-trivial 2×2 nilpotent matrix has module

K[T ]/T 2 , showing algebraic multiplicity 2 but geometric multiplicity 1. In more detail: • the eigenvalues (with geometric multiplicity) of the operator correspond to the (reduced) points of the variety, with multiplicity; • the primary decomposition of the module corresponds to the unreduced points of the variety; • a diagonalizable (semisimple) operator corresponds to a reduced variety; • a cyclic module (one generator) corresponds to the operator having a cyclic vector (a vector whose orbit under T spans the space); • the last invariant factor of the module equals the minimal polynomial of the operator, and the product of the invariant factors equals the characteristic polynomial.

800

CHAPTER 264. SPECTRUM OF A RING

264.8 Generalizations The spectrum can be generalized from rings to C*-algebras in operator theory, yielding the notion of the spectrum of a C*-algebra. Notably, for a Hausdorff space, the algebra of scalars (the bounded continuous functions on the space, being analogous to regular functions) are a commutative C*-algebra, with the space being recovered as a topological space from MSpec of the algebra of scalars, indeed functorially so; this is the content of the Banach–Stone theorem. Indeed, any commutative C*-algebra can be realized as the algebra of scalars of a Hausdorff space in this way, yielding the same correspondence as between a ring and its spectrum. Generalizing to non-commutative C*-algebras yields noncommutative topology.

264.9 See also • scheme • projective scheme • Spectrum of a matrix • Constructible topology • Serre’s theorem on affineness

264.10 References • Cox, David; O'Shea, Donal; Little, John (1997), Ideals, Varieties, and Algorithms, Berlin, New York: SpringerVerlag, ISBN 978-0-387-94680-1 • Eisenbud, David; Harris, Joe (2000), The geometry of schemes, Graduate Texts in Mathematics 197, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98637-1, MR 1730819 • Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-902449, MR 0463157

264.11 External links • Kevin R. Coombes: The Spectrum of a Ring • Miles Reid, Undergraduate Commutative Algebra, page 22

Chapter 265

Square-free In mathematics, an element r of a unique factorization domain R is called square-free if it is not divisible by a non-trivial square. That is, every s such that s2 | r is a unit of R.

265.1 Alternate characterizaions Square-free elements may be also characterized using their prime decomposition. The unique factorization property means that a non-zero non-unit r can be represented as a product of prime elements

r = p1 p2 · · · pn Then r is square-free if and only if the primes pi are pairwise non-associated (i.e. that it doesn't have two of the same prime as factors, which would make it divisible by a square number).

265.2 Examples Common examples of square-free elements include square-free integers and square-free polynomials.

265.3 See also • Prime number

265.4 References • David Darling (2004) The Universal Book of Mathematics: From Abracadabra to Zeno’s Paradoxes John Wiley & Sons • Baker, R. C. “The square-free divisor problem.” The Quarterly Journal of Mathematics 45.3 (1994): 269-277.

801

Chapter 266

Stably finite ring In algebra, a ring R is said to be stably finite (or weakly finite) if, for all square matrices A, B of the same size over R, AB = 1 implies BA = 1. This is a slightly stronger property for a ring than its having the invariant basis number: any nontrivial[1] stably finite ring has IBN. Commutative rings, noetherian rings and artinian rings are stably finite. A subring of a stably finite ring and a matrix ring over a stably finite ring is stably finite. A ring satisfying Klein’s nilpotence condition is stably finite.

266.1 References [1] A trivial ring is stably finite but doesn't have IBN.

• P.M. Cohn (2003). Basic Algebra, Springer.

802

Chapter 267

Stably free module In mathematics, a stably free module is a module which is close to being free.

267.1 Definition A finitely generated module M over a ring R is stably free if there exist free finitely generated modules F and G over R such that

M ⊕ F = G.

267.2 Properties • A projective module is stably free if and only if it possesses a finite free resolution (Theorem XXI.2.1 of Lang’s Algebra). • An infinitely generated module is stably free if and only if it is free.

267.3 See also • Free object • Eilenberg–Mazur swindle • Hermite ring

267.4 References • Page 840 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-201-55540-0, Zbl 0848.13001

803

Chapter 268

Structure theorem for finitely generated modules over a principal ideal domain In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules can be uniquely decomposed in much the same way that integers have a prime factorization. The result provides a simple framework to understand various canonical form results for square matrices over fields.

268.1 Statement When a vector space over a field F has a finite generating set, then one may extract from it a basis consisting of a finite number n of vectors, and the space is therefore isomorphic to Fn . The corresponding statement with the F generalized to a principal ideal domain R is no longer true, as a finitely generated module over R need not have any basis. However such a module is still isomorphic to a quotient of some module Rn with n finite (to see this it suffices to construct the morphism that sends the elements of the canonical basis Rn to the generators of the module, and take the quotient by its kernel.) By changing the choice of generating set, one can in fact describe the module as the quotient of some Rn by a particularly simple submodule, and this is the structure theorem. The structure theorem for finitely generated modules over a principal ideal domain usually appears in the following two forms.

268.1.1

Invariant factor decomposition

For every finitely generated module M over a principal ideal domain R, there is a unique decreasing sequence of proper ideals (d1 ) ⊇ (d2 ) ⊇ · · · ⊇ (dn ) such that M isomorphic to the sum of cyclic modules: M∼ =

⊕

R/(di ) = R/(d1 ) ⊕ R/(d2 ) ⊕ · · · ⊕ R/(dn ).

i

The generators di of the ideals are unique up to multiplication by a unit, and are called invariant factors of M. Since the ideals should be proper, these factors must not themselves be invertible (this avoids trivial factors in the sum), and the inclusion of the ideals means one has divisibility d1 | d2 | · · · | dn . The free part is visible in the part of the decomposition corresponding to factors di = 0 . Such factors, if any, occur at the end of the sequence. While the direct sum is uniquely determined by M, the isomorphism giving the decomposition itself is not unique in general. For instance if R is actually a field, then all occurring ideals must be zero, and one obtains the decomposition of a finite dimensional vector space into a direct sum of one-dimensional subspaces; the number of such factors is fixed, namely the dimension of the space, but there is in a lot of freedom for choosing the subspaces themselves (if dim M > 1). 804

268.2. PROOFS

805

The nonzero di elements, together with the number of di which are zero, form a complete set of invariants for the module. Explicitly, this means that any two modules sharing the same set of invariants are necessarily isomorphic. Some prefer to write the free part of M separately:

Rf ⊕

⊕

R/(di ) = Rf ⊕ R/(d1 ) ⊕ R/(d2 ) ⊕ · · · ⊕ R/(dn−f )

i

where the visible di are nonzero, and f is the number of di 's in the original sequence which are 0.

268.1.2

Primary decomposition

Every finitely generated module M over a principal ideal domain R is isomorphic to one of the form ⊕ R/(qi ) i

where (qi ) ̸= R and the (qi ) are primary ideals. The qi are unique (up to multiplication by units). The elements qi are called the elementary divisors of M. In a PID, nonzero primary ideals are powers of primes, and so (qi ) = (pri i ) = (pi )ri . When qi = 0 , the resulting indecomposable module is R itself, and this is inside the part of M that is a free module. The summands R/(qi ) are indecomposable, so the primary decomposition is a decomposition into indecomposable modules, and thus every finitely generated module over a PID is a completely decomposable module. Since PID’s are Noetherian rings, this can be seen as a manifestation of the Lasker-Noether theorem. As before, it is possible to write the free part (where qi = 0 ) separately and express M as:

Rf ⊕ (

⊕

R/(qi ))

i

where the visible qi are nonzero.

268.2 Proofs One proof proceeds as follows: • Every finitely generated module over a PID is also finitely presented because a PID is Noetherian, an even stronger condition than coherence. • Take a presentation, which is a map Rr → Rg (relations to generators), and put it in Smith normal form. This yields the invariant factor decomposition, and the diagonal entries of Smith normal form are the invariant factors. Another outline of a proof: • Denote by tM the torsion submodule of M. Then M/tM is a finitely generated torsion free module, and such a module over a commutative PID is a free module of finite rank, so it is isomorphic to Rn for a positive integer n. This free module can be embedded as a submodule F of M, such that the embedding splits (is a right inverse of) the projection map; it suffices to lift each of the generators of F into M. As a consequence M = tM ⊕ F . • For a prime p in R we can then speak of Np = {m ∈ tM | ∃i, mpi = 0} for each prime p. This is a submodule of tM, and it turns out that each Np is a direct sum of cyclic modules, and that tM is a direct sum of Np for a finite number of distinct primes p. • Putting the previous two steps together, M is decomposed into cyclic modules of the indicated types.

806CHAPTER 268. STRUCTURE THEOREM FOR FINITELY GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN

268.3 Corollaries This includes the classification of finite-dimensional vector spaces as a special case, where R = K . Since fields have no non-trivial ideals, every finitely generated vector space is free. Taking R = Z yields the fundamental theorem of finitely generated abelian groups. Let T be a linear operator on a finite-dimensional vector space V over K. Taking R = K[T ] , the algebra of polynomials with coefficients in K evaluated at T, yields structure information about T. V can be viewed as a finitely generated module over K[T ] . The last invariant factor is the minimal polynomial, and the product of invariant factors is the characteristic polynomial. Combined with a standard matrix form for K[T ]/p(T ) , this yields various canonical forms: • invariant factors + companion matrix yields Frobenius normal form (aka, rational canonical form) • primary decomposition + companion matrix yields primary rational canonical form • primary decomposition + Jordan blocks yields Jordan canonical form (this latter only holds over an algebraically closed field)

268.4 Uniqueness While the invariants (rank, invariant factors, and elementary divisors) are unique, the isomorphism between M and its canonical form is not unique, and does not even preserve the direct sum decomposition. This follows because there are non-trivial automorphisms of these modules which do not preserve the summands. However, one has a canonical torsion submodule T, and similar canonical submodules corresponding to each (distinct) invariant factor, which yield a canonical sequence:

0 < · · · < T < M. Compare composition series in Jordan–Hölder theorem. For instance,[if M ]≈ Z ⊕ Z/2 , and (1, 0), (0, 1) is one basis, then (1, 1), (0, 1) is another basis, and the change of 1 1 basis matrix does not preserve the summand Z . However, it does preserve the Z/2 summand, as this is the 0 1 torsion submodule (equivalently here, the 2-torsion elements).

268.5 Generalizations 268.5.1

Groups

The Jordan–Hölder theorem is a more general result for finite groups (or modules over an arbitrary ring). In this generality, one obtains a composition series, rather than a direct sum. The Krull–Schmidt theorem and related results give conditions under which a module has something like a primary decomposition, a decomposition as a direct sum of indecomposable modules in which the summands are unique up to order.

268.5.2

Primary decomposition

The primary decomposition generalizes to finitely generated modules over commutative Noetherian rings, and this result is called the Lasker–Noether theorem.

268.6. REFERENCES

268.5.3

807

Indecomposable modules

By contrast, unique decomposition into indecomposable submodules does not generalize as far, and the failure is measured by the ideal class group, which vanishes for PIDs. For rings that are not principal ideal domains, unique decomposition need not even hold for modules over a ring generated by two elements. For the ring R = Z[√−5], both the module R and its submodule M generated by 2 and 1 + √−5 are indecomposable. While R is not isomorphic to M, R ⊕ R is isomorphic to M ⊕ M; thus the images of the M summands give indecomposable submodules L1 , L2 < R ⊕ R which give a different decomposition of R ⊕ R. The failure of uniquely factorizing R ⊕ R into a direct sum of indecomposable modules is directly related (via the ideal class group) to the failure of the unique factorization of elements of R into irreducible elements of R.

268.5.4

Non-finitely generated modules

Similarly for modules that are not finitely generated, one cannot expect such a nice decomposition: even the number of factors may vary. There are Z-submodules of Q4 which are simultaneously direct sums of two indecomposable modules and direct sums of three indecomposable modules, showing the analogue of the primary decomposition cannot hold for infinitely generated modules, even over the integers, Z. Another issue that arises with non-finitely generated modules is that there are torsion-free modules which are not free. For instance, consider the ring Z of integers. Then Q is a torsion-free Z-module which is not free. Another classical example of such a module is the Baer–Specker group, the group of all sequences of integers under termwise addition. In general, the question of which infinitely generated torsion-free abelian groups are free depends on which large cardinals exist. A consequence is that any structure theorem for infinitely generated modules depends on a choice of set theory axioms and may be invalid under a different choice.

268.6 References • Dummit, David S.; Foote, Richard M. (2004), Abstract algebra (3rd ed.), New York: Wiley, ISBN 978-0-47143334-7, MR 2286236 • Hungerford, Thomas W. (1980), Algebra, New York: Springer, pp. 218–226, Section IV.6: Modules over a Principal Ideal Domain, ISBN 978-0-387-90518-1 • Jacobson, Nathan (1985), Basic algebra. I (2 ed.), New York: W. H. Freeman and Company, pp. xviii+499, ISBN 0-7167-1480-9, MR 780184 • Lam, T. Y. (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Springer-Verlag, ISBN 978-0-387-98428-5 • Adhikari, M.R.; Adhikari, Avishek (2014). Basic Modern Algebra with Applications. New Delhi, New York: Springer. p. 637. ISBN 978-81-322-1598-1.

Chapter 269

Subring In mathematics, a subring of R is a subset of a ring that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and which shares the same multiplicative identity as R. For those who define rings without requiring the existence of a multiplicative identity, a subring of R is just a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R). The latter gives a strictly weaker condition, even for rings that do have a multiplicative identity, so that for instance all ideals become subrings (and they may have a multiplicative identity that differs from the one of R). With definition requiring a multiplicative identity (which is used in this article), the only ideal of R that is a subring of R is R itself.

269.1 Formal definition A subring of a ring (R, +, ∗, 0, 1) is a subset S of R that preserves the structure of the ring, i.e. a ring (S, +, ∗, 0, 1) with S ⊆ R. Equivalently, it is both a subgroup of (R, +, 0) and a submonoid of (R, ∗, 1).

269.2 Examples The ring Z and its quotients Z/nZ have no subrings (with multiplicative identity) other than the full ring. Every ring has a unique smallest subring, isomorphic to some ring Z/nZ with n a nonnegative integer (see characteristic). The integers Z correspond to n = 0 in this statement, since Z is isomorphic to Z/0Z.

269.3 Subring test The subring test is a theorem that states that for any ring R, a subset of R is a subring if it is closed under multiplication and subtraction, and contains the multiplicative identity of R. As an example, the ring Z of integers is a subring of the field of real numbers and also a subring of the ring of polynomials Z[X].

269.4 Ring extensions Not to be confused with a ring-theoretic analog of a group extension. If S is a subring of a ring R, then equivalently R is said to be a ring extension of S, written as R/S in similar notation to that for field extensions. 808

269.5. SUBRING GENERATED BY A SET

809

269.5 Subring generated by a set Let R be a ring. Any intersection of subrings of R is again a subring of R. Therefore, if X is any subset of R, the intersection of all subrings of R containing X is a subring S of R. S is the smallest subring of R containing X. (“Smallest” means that if T is any other subring of R containing X, then S is contained in T.) S is said to be the subring of R generated by X. If S = R, we may say that the ring R is generated by X.

269.6 Relation to ideals Proper ideals are subrings that are closed under both left and right multiplication by elements from R. If one omits the requirement that rings have a unity element, then subrings need only be non-empty and otherwise conform to the ring structure, and ideals become subrings. Ideals may or may not have their own multiplicative identity (distinct from the identity of the ring): • The ideal I = {(z,0) | z in Z} of the ring Z × Z = {(x,y) | x,y in Z} with componentwise addition and multiplication has the identity (1,0), which is different from the identity (1,1) of the ring. So I is a ring with unity, and a “subring-without-unity”, but not a “subring-with-unity” of Z × Z. • The proper ideals of Z have no multiplicative identity. If I is a prime ideal of a commutative ring R, then the intersection of I with any subring S of R remains prime in S. In this case one says that I lies over I ∩ S. The situation is more complicated when R is not commutative.

269.7 Profile by commutative subrings A ring may be profiled by the variety of commutative subrings that it hosts: • The quaternion ring H contains only the complex plane as a planar subring • The coquaternion ring contains three types of commutative planar subrings: the dual number plane, the splitcomplex number plane, as well as the ordinary complex plane • The ring of 3 × 3 real matrices also contains 3-dimensional commutative subrings generated by the identity matrix and a nilpotent ε of order 3 (εεε = 0 ≠ εε). For instance, the Heisenberg group can be realized as the join of the groups of units of two of these nilpotent-generated subrings of 3 × 3 matrices.

269.8 See also • Integral extension • Group extension • Algebraic extension • Ore extension

269.9 References • Iain T. Adamson (1972). Elementary rings and modules. University Mathematical Texts. Oliver and Boyd. pp. 14–16. ISBN 0-05-002192-3. • Page 84 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 9780-201-55540-0, Zbl 0848.13001 • David Sharpe (1987). Rings and factorization. Cambridge University Press. pp. 15–17. ISBN 0-521-33718-6.

Chapter 270

Supermodule In mathematics, a supermodule is a Z2 -graded module over a superring or superalgebra. Supermodules arise in super linear algebra which is a mathematical framework for studying the concept supersymmetry in theoretical physics. Supermodules over a commutative superalgebra can be viewed as generalizations of super vector spaces over a (purely even) field K. Supermodules often play a more prominent role in super linear algebra than do super vector spaces. These reason is that it is often necessary or useful to extend the field of scalars to include odd variables. In doing so one moves from fields to commutative superalgebras and from vector spaces to modules. In this article, all superalgebras are assumed be associative and unital unless stated otherwise.

270.1 Formal definition Let A be a fixed superalgebra. A right supermodule over A is a right module E over A with a direct sum decomposition (as an abelian group) E = E0 ⊕ E1 such that multiplication by elements of A satisfies Ei Aj ⊆ Ei+j for all i and j in Z2 . The subgroups Ei are then right A0 -modules. The elements of Ei are said to be homogeneous. The parity of a homogeneous element x, denoted by |x|, is 0 or 1 according to whether it is in E 0 or E 1 . Elements of parity 0 are said to be even and those of parity 1 to be odd. If a is a homogeneous scalar and x is a homogeneous element of E then |x·a| is homogeneous and |x·a| = |x| + |a|. Likewise, left supermodules and superbimodules are defined as left modules or bimodules over A whose scalar multiplications respect the gradings in the obvious manner. If A is supercommutative, then every left or right supermodule over A may be regarded as a superbimodule by setting a · x = (−1)|a||x| x · a for homogeneous elements a ∈ A and x ∈ E, and extending by linearity. If A is purely even this reduces to the ordinary definition.

270.2 Homomorphisms A homomorphism between supermodules is a module homomorphism that preserves the grading. Let E and F be right supermodules over A. A map 810

270.3. REFERENCES

811

ϕ:E→F is a supermodule homomorphism if • ϕ(x + y) = ϕ(x) + ϕ(y) • ϕ(x · a) = ϕ(x) · a • ϕ(Ei ) ⊆ Fi for all a∈A and all x,y∈E. The set of all module homomorphisms from E to F is denoted by Hom(E, F). In many cases, it is necessary or convenient to consider a larger class of morphisms between supermodules. Let A be a supercommutative algebra. Then all supermodules over A be regarded as superbimodules in a natural fashion. For supermodules E and F, let Hom(E, F) denote the space of all right A-linear maps (i.e. all module homomorphisms from E to F considered as ungraded right A-modules). There is a natural grading on Hom(E, F) where the even homomorphisms are those that preserve the grading ϕ(Ei ) ⊆ Fi and the odd homomorphisms are those that reverse the grading ϕ(Ei ) ⊆ F1−i . If φ ∈ Hom(E, F) and a ∈ A are homogeneous then ϕ(x · a) = ϕ(x) · a

ϕ(a · x) = (−1)|a||ϕ| a · ϕ(x).

That is, the even homomorphisms are both right and left linear whereas the odd homomorphism are right linear but left antilinear (with respect to the grading automorphism). The set Hom(E, F) can be given the structure of a bimodule over A by setting (a · ϕ)(x) = a · ϕ(x) (ϕ · a)(x) = ϕ(a · x). With the above grading Hom(E, F) becomes a supermodule over A whose even part is the set of all ordinary supermodule homomorphisms

Hom0 (E, F ) = Hom(E, F ). In the language of category theory, the class of all supermodules over A forms a category with supermodule homomorphisms as the morphisms. This category is a symmetric monoidal closed category under the super tensor product whose internal Hom functor is given by Hom.

270.3 References • Deligne, Pierre; John W. Morgan (1999). “Notes on Supersymmetry (following Joseph Bernstein)". Quantum Fields and Strings: A Course for Mathematicians 1. American Mathematical Society. pp. 41–97. ISBN 0-8218-2012-5. • Manin, Y. I. (1997). Gauge Field Theory and Complex Geometry ((2nd ed.) ed.). Berlin: Springer. ISBN 3-540-61378-1. • Varadarajan, V. S. (2004). Supersymmetry for Mathematicians: An Introduction. Courant Lecture Notes in Mathematics 11. American Mathematical Society. ISBN 0-8218-3574-2.

Chapter 271

Support of a module In commutative algebra, the support of a module M over a commutative ring A is the set of all prime ideals p of A such that Mp ̸= 0 .[1] It is denoted by Supp(M ) . In particular, M = 0 if and only if its support is empty. • Let 0 → M ′ → M → M ′′ → 0 be an exact sequence of A-modules. Then Supp(M ) = Supp(M ′ ) ∪ Supp(M ′′ ). • If M is a sum of submodules Mλ , then Supp(M ) = ∪λ supp(Mλ ). • If M is a finitely generated A-module, then Supp(M ) is the set of all prime ideals containing the annihilator of M. In particular, it is closed. • If M, N are finitely generated A-modules, then Supp(M ⊗A N ) = Supp(M ) ∩ Supp(N ). • If M is a finitely generated A-module and I is an ideal of A, then Supp(M /IM ) is the set of all prime ideals containing I + Ann(M ). This is V (I) ∩ Supp(M ) .

271.1 See also • Associated prime

271.2 References [1] EGA 0I, 1.7.1.

• Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas”. Publications Mathématiques de l'IHÉS 4. doi:10.1007/bf02684778. MR 0217083.

812

Chapter 272

Sylvester domain In mathematics, a Sylvester domain, named after James Joseph Sylvester by Dicks & Sontag (1978), is a ring in which Sylvester’s law of nullity holds. This means that if A is an m by n matrix and B an n by s matrix over R, then ρ(AB) ≥ ρ(A) + ρ(B) – n where ρ is the inner rank of a matrix. The inner rank of an m by n matrix is the smallest integer r such that the matrix is a product of an m by r matrix and an r by n matrix. Sylvester (1884) showed that fields satisfy Sylvester’s law of nullity and are therefore Sylvester domains.

272.1 References • Dicks, Warren; Sontag, Eduardo D. (1978), “Sylvester domains”, Journal of Pure and Applied Algebra 13 (3): 243–275, doi:10.1016/0022-4049(78)90011-7, ISSN 0022-4049, MR 509164 • Sylvester, James Joseph (1884), “On involutants and other allied species of invariants to matrix systems”, Johns Hopkins university circulars III: 9–12, 34–35, Reprinted in collected papers volume IV, paper 15

813

Chapter 273

Symmetric algebra Not to be confused with symmetric tensors. In mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is the free commutative unital associative algebra over K containing V. It corresponds to polynomials with indeterminates in V, without choosing coordinates. The dual, S(V ∗ ) corresponds to polynomials on V. A Frobenius algebra whose bilinear form is symmetric is also called a symmetric algebra, but is not discussed here.

273.1 Construction It turns out that S(V) is in effect the same as the polynomial ring, over K, in indeterminates that are basis vectors for V. Therefore this construction only brings something extra when the “naturality” of looking at polynomials this way has some advantage. It is possible to use the tensor algebra T(V) to describe the symmetric algebra S(V). In fact we pass from the tensor algebra to the symmetric algebra by forcing it to be commutative; if elements of V commute, then tensors in them must, so that we construct the symmetric algebra from the tensor algebra by taking the quotient algebra of T(V) by the ideal generated by all differences of products v ⊗ w − w ⊗ v. for v and w in V.

273.1.1

Grading

Just as with a polynomial ring, there is a direct sum decomposition of S(V) as a graded algebra, into summands Sk (V) which consist of the linear span of the monomials in vectors of V of degree k, for k = 0, 1, 2, ... (with S 0 (V) = K and S 1 (V) = V). The K-vector space Sk (V) is the k-th symmetric power of V. (The case k = 2, for example, is the symmetric square and denoted Sym2 (V).) It has a universal property with respect to symmetric multilinear operators defined on V k . In terms of the tensor algebra grading, Sk (V) is the quotient space of Tk (V) by the subspace generated by all differences of products v ⊗ w − w ⊗ v. and products of these with other algebra elements. 814

273.2. INTERPRETATION AS POLYNOMIALS

273.1.2

815

Distinction with symmetric tensors

The symmetric algebra and symmetric tensors are easily confused: the symmetric algebra is a quotient of the tensor algebra, while the symmetric tensors are a subspace of the tensor algebra. The symmetric algebra must be a quotient to satisfy its universal property (since every symmetric algebra is an algebra, the tensor algebra maps to the symmetric algebra). Conversely, symmetric tensors are defined as invariants: given the natural action of the symmetric group on the tensor algebra, the symmetric tensors are the subspace on which the symmetric group acts trivially. Note that under the tensor product, symmetric tensors are not a subalgebra: given vectors v and w, they are trivially symmetric 1-tensors, but v ⊗ w is not a symmetric 2-tensor. The grade 2 part of this distinction is the difference between symmetric bilinear forms (symmetric 2-tensors) and quadratic forms (elements of the symmetric square), as described in ε-quadratic forms. In characteristic 0 symmetric tensors and the symmetric algebra can be identified. In any characteristic, there is a symmetrization map from the symmetric algebra to the symmetric tensors, given by:

v1 · · · vk 7→

∑

vσ(1) ⊗ · · · ⊗ vσ(k) .

σ∈Sk

The composition with the inclusion of the symmetric tensors in the tensor algebra and the quotient to the symmetric algebra is multiplication by k! on the kth graded component. Thus in characteristic 0, the symmetrization map is an isomorphism of graded vector spaces, and one can identify symmetric tensors with elements of the symmetric algebra. One divides by k! to make this a section of the quotient map:

v1 · · · vk 7→

1 ∑ vσ(1) ⊗ · · · ⊗ vσ(k) . k! σ∈Sk

For instance, vw 7→ 21 (v ⊗ w + w ⊗ v) . This is related to the representation theory of the symmetric group: in characteristic 0, over an algebraically closed field, the group algebra is semisimple, so every representation splits into a direct sum of irreducible representations, and if T = S ⊕V, one can identify S as either a subspace of T or as the quotient T/V.

273.2 Interpretation as polynomials Main article: Ring of polynomial functions Given a vector space V, the polynomials on this space are S(V ∗ ), the symmetric algebra of the dual space: a polynomial on a space evaluates vectors on the space, via the pairing S(V ∗ ) × V → K . For instance, given the plane with a basis {(1,0), (0,1)}, the (homogeneous) linear polynomials on K 2 are generated by the coordinate functionals x and y. These coordinates are covectors: given a vector, they evaluate to their coordinate, for instance:

x(2, 3) = 2, and y(2, 3) = 3. Given monomials of higher degree, these are elements of various symmetric powers, and a general polynomial is an element of the symmetric algebra. Without a choice of basis for the vector space, the same holds, but one has a polynomial algebra without choice of basis. Conversely, the symmetric algebra of the vector space itself can be interpreted, not as polynomials on the vector space (since one cannot evaluate an element of the symmetric algebra of a vector space against a vector in that space: there is no pairing between S(V) and V), but polynomials in the vectors, such as v2 − vw + uv.

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273.2.1

CHAPTER 273. SYMMETRIC ALGEBRA

Symmetric algebra of an affine space

One can analogously construct the symmetric algebra on an affine space (or its dual, which corresponds to polynomials on that affine space). The key difference is that the symmetric algebra of an affine space is not a graded algebra, but a filtered algebra: one can determine the degree of a polynomial on an affine space, but not its homogeneous parts. For instance, given a linear polynomial on a vector space, one can determine its constant part by evaluating at 0. On an affine space, there is no distinguished point, so one cannot do this (choosing a point turns an affine space into a vector space).

273.3 Categorical properties The symmetric algebra on a vector space is a free object in the category of commutative unital associative algebras (in the sequel, “commutative algebras”). Formally, the map that sends a vector space to its symmetric algebra is a functor from vector spaces over K to commutative algebras over K, and is a free object, meaning that it is left adjoint to the forgetful functor that sends a commutative algebra to its underlying vector space. The unit of the adjunction is the map V → S(V) that embeds a vector space in its symmetric algebra. Commutative algebras are a reflective subcategory of algebras; given an algebra A, one can quotient out by its commutator ideal generated by ab – ba, obtaining a commutative algebra, analogously to abelianization of a group. The construction of the symmetric algebra as a quotient of the tensor algebra is, as functors, a composition of the free algebra functor with this reflection.

273.4 Analogy with exterior algebra The S k are functors comparable to the exterior powers; here, though, the dimension grows with k; it is given by ( dim(S k (V )) =

) n+k−1 k

where n is the dimension of V. This binomial coefficient is the number of n-variable monomials of degree k.

273.5 Module analog The construction of the symmetric algebra generalizes to the symmetric algebra S(M) of a module M over a commutative ring. If M is a free module over the ring R, then its symmetric algebra is isomorphic to the polynomial algebra over R whose indeterminates are a basis of M, just like the symmetric algebra of a vector space. However, that is not true if M is not free; then S(M) is more complicated.

273.6 As a universal enveloping algebra The symmetric algebra S(V) is the universal enveloping algebra of an abelian Lie algebra, i.e. one in which the Lie bracket is identically 0.

273.7 See also • exterior algebra, the anti-symmetric analog • Weyl algebra, a quantum deformation of the symmetric algebra by a symplectic form • Clifford algebra, a quantum deformation of the exterior algebra by a quadratic form

273.8. REFERENCES

273.8 References • Bourbaki, Nicolas (1989), Elements of mathematics, Algebra I, Springer-Verlag, ISBN 3-540-64243-9

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Chapter 274

Syzygy (mathematics) For other uses, see Syzygy (disambiguation). In mathematics, a syzygy (from Greek συζυγία 'pair') is a relation between the generators of a module M. The set of all such relations is called the “first syzygy module of M". A relation between generators of the first syzygy module is called a “second syzygy” of M, and the set of all such relations is called the “second syzygy module of M". Continuing in this way, we derive the nth syzygy module of M by taking the set of all relations between generators of the (n − 1)th syzygy module of M. If M is finitely generated over a polynomial ring over a field, this process terminates after a finite number of steps; i.e., eventually there will be no more syzygies (see Hilbert’s syzygy theorem). The syzygy modules of M are not unique, for they depend on the choice of generators at each step. The sequence of the successive syzygy modules of a module M is the sequence of the successive images (or kernels) in a free resolution of this module. Buchberger’s algorithm for computing Gröbner bases allows to compute the first syzygy module: The reduction to zero of the S-polynomial of a pair of polynomials in a Gröbner basis provides a syzygy, and these syzygies generate the first module of syzygies.

274.1 Further reading • Hazewinkel, Michiel, ed. (2001), “Syzygy”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-0104 • Wiegand, Roger (April 2006), “WHAT IS...a Syzygy?", Notices of the American Mathematical Society 53 (4): 456–457, retrieved 23 May 2011

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Chapter 275

Tensor product of algebras In mathematics, the tensor product of two R-algebras is also an R-algebra. This gives us a tensor product of algebras. The special case R = Z gives us a tensor product of rings, since rings may be regarded as Z-algebras. Let R be a commutative ring and let A and B be R-algebras. Since A and B may both be regarded as R-modules, we may form their tensor product

A ⊗R B which is also an R-module. We can give the tensor product the structure of an algebra by defining[1]

(a1 ⊗ b1 )(a2 ⊗ b2 ) = a1 a2 ⊗ b1 b2 and then extending by linearity to all of A ⊗R B. This product is easily seen to be R-bilinear, associative, and unital with an identity element given by 1A ⊗ 1B,[2] where 1A and 1B are the identities of A and B. If A and B are both commutative then the tensor product is commutative as well. The tensor product turns the category of all R-algebras into a symmetric monoidal category. There are natural homomorphisms of A and B to A ⊗R B given by[3]

a 7→ a ⊗ 1B b 7→ 1A ⊗ b These maps make the tensor product a coproduct in the category of commutative R-algebras. The tensor product is not the coproduct in the category of all R-algebras. There the coproduct is given by a more general free product of algebras. Nevertheless the tensor product of non-commutative algebras can be described by an universal property similar to that of the coproduct: Hom(A ⊗ B, X) ∼ = {(f, g) ∈ Hom(A, X) × Hom(B, X) | ∀a ∈ A, b ∈ B : [f (a), g(b)] = 0} The natural isomorphism is given by identifying a morphism ϕ : A ⊗ B → X on the left hand side with the pair of morphism (f, g) on the right hand side where f (a) := ϕ(a ⊗ 1) and similarly g(b) := ϕ(1 ⊗ b) . The tensor product of algebras is of constant use in algebraic geometry: working in the opposite category to that of commutative R-algebras, it provides pullbacks of affine schemes, otherwise known as fiber products.

275.1 See also • Extension of scalars 819

820

CHAPTER 275. TENSOR PRODUCT OF ALGEBRAS

• Tensor product of modules • Tensor product of fields • Linearly disjoint • Multilinear subspace learning

275.2 Notes [1] Kassel (1995), p. 32. [2] Kassel (1995), p. 32. [3] Kassel (1995), p. 32.

275.3 References • Kassel, Christian (1995), Quantum groups, Graduate texts in mathematics 155, Springer, ISBN 978-0-38794370-1.

Chapter 276

Tensor product of modules In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps (module homomorphisms). The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology and algebraic geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear operations via linear operations. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the module in a universal way.

276.1 Balanced product For a ring R, a right R-module M, a left R-module N, and an abelian group G, a map φ: M × N → G is said to be R-balanced, R-middle-linear or an R-balanced product if for all m, m′ in M, n, n′ in N, and r in R the following hold:

The set of all such balanced products over R from M × N to G is denoted by LR(M, N; G). If φ, ψ are balanced products, then the operations φ + ψ and −φ defined pointwise are each a balanced product. This turns the set LR(M, N; G) into an abelian group. For M and N fixed, the map G ↦ LR(M, N; G) is a functor from the category of abelian groups to the category of sets. The morphism part is given by mapping a group homomorphism g : G → G′ to the function φ ↦ g ∘ φ, which goes from LR(M, N; G) to LR(M, N; G′). Remarks 1. Property (Dl) states the left and property (Dr) the right distributivity of φ over addition. 2. Property (A) resembles some associative property of φ. 3. Every ring R is an R-R-bimodule. So the ring multiplication (r, r′) ↦ r ⋅ r′ in R is an R-balanced product R × R → R.

276.2 Definition For a ring R, a right R-module M, a left R-module N, the tensor product over R 821

822

CHAPTER 276. TENSOR PRODUCT OF MODULES

M ⊗R N is an abelian group together with a balanced product (as defined above)

⊗ : M × N → M ⊗R N which is universal in the following sense:[1]

× M::

N:::::::::::::::::M:::::N R ~

f:::::::::::: f

G For every abelian group G and every balanced product f :M ×N →G there is a unique group homomorphism f˜ : M ⊗R N → G such that f˜ ◦ ⊗ = f. As with all universal properties, the above property defines the tensor product uniquely up to a unique isomorphism: any other object and balanced product with the same properties will be isomorphic to M ⊗R N and ⊗. Indeed, the mapping ⊗ is called canonical, or more explicitly: the canonical mapping (or balanced product) of the tensor product.[2]

276.2. DEFINITION

823

The definition does not prove the existence of M ⊗R N; see below for a construction. The tensor product can also be defined as a representing object for the functor G → LR(M,N;G); explicitly, this means there is a natural isomorphism: HomZ (M ⊗R N, G) ≃ LR (M, N ; G), g 7→ g ◦ ⊗. This is a succinct way of stating the universal mapping property given above. (A priori, if one is given this is natural isomorphism, then ⊗ can be recovered by taking G = M ⊗R N and then mapping the identity map.) Similarly, given the natural identification LR (M, N ; G) = HomR (M, HomZ (N, G)) ,[3] one can also define M ⊗R N by the formula HomZ (M ⊗R N, G) ≃ HomR (M, HomZ (N, G)) This is known as the tensor-hom adjunction; see also § Properties. For each x in M, y in N, one writes x⊗y for the image of (x, y) under the canonical map ⊗ : M × N → M ⊗R N . It is often called a pure tensor. Strictly speaking, the correct notation would be x ⊗R y but it is conventional to drop R here. Then, immediately from the definition, there are relations:

The universal property of a tensor product has the following important consequence: Proposition — Every element of M ⊗R N can be written, non-uniquely, as ∑

xi ⊗ yi , xi ∈ M, yi ∈ N.

i

In other words, the image of ⊗ generates M ⊗R N . Furthermore, if f is a function defined on elements x ⊗ y with values in an abelian group G, then f extends uniquely to the homomorphism defined on the whole M ⊗R N if and only if f (x ⊗ y) is Z-bilinear in x and y. Proof: For the first statement, let L be the subgroup of M ⊗R N generated by elements of the form in question, Q = (M ⊗R N )/L and q the quotient map to Q. We have: 0 = q ◦ ⊗ as well as 0 = 0 ◦ ⊗ . Hence, by the uniqueness part of the universal property, q = 0. The second statement is because to define a module homomorphism, it is enough to define it on the generating set of the module. □ The proposition says that one can work with explicit elements of the tensor products instead of invoking the universal property directly each time. This is very convenient in practice. For example, if R is commutative, then M ⊗R N can naturally be furnished with the R-scalar multiplication by extending r · (x ⊗ y) := rx ⊗ y = x ⊗ ry to the whole M ⊗R N by the previous proposition (strictly speaking, what is needed is a bimodule structure not commutativity; see a paragraph below). Equipped with this R-module structure, M ⊗R N satisfies a universal property similar to the above: for any R-module G, there is a natural isomorphism: HomR (M ⊗R N, G) ≃ { R-bilinear maps from M × N to G}, g 7→ g ◦ ⊗ . If R is not necessarily commutative but if M has a left action by a ring S (for example, R), then M ⊗R N can be given the left S-module structure, like above, by the formula s · (x ⊗ y) := sx ⊗ y If N has a right action by a ring S, then, in the analogous way, M ⊗R N becomes a right S-module.

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CHAPTER 276. TENSOR PRODUCT OF MODULES

276.2.1

Tensor product of linear maps and a change of base ring

Given linear maps f : M → M ′ of right modules over a ring R and g : N → N ′ of left modules, there is a unique group homomorphism

f ⊗ g : M ⊗R N → M ′ ⊗R N ′ such that (f ⊗ g)(x ⊗ y) = f (x) ⊗ g(y) . The construction has a consequence that tensoring is a functor: each right R-module M determines the functor

M ⊗R − : R − Mod → Ab from the category of left modules to the category of abelian groups that sends N to M ⊗ N and a module homomorphism f to the group homomorphism 1 ⊗ f. If f : R → S is a ring homomorphism and if M is a right S-module and N a left S-module, then there is the canonical surjective homomorphism:

M ⊗R N → M ⊗S N ⊗

S induced by M × N →M ⊗S N .[4] (The resulting map is surjective since pure tensors x ⊗ y generate the whole module.) In particular, taking R to be Z, this shows every tensor product of modules is a quotient of a tensor product of abelian groups.

See also: Tensor product § Tensor product of linear maps.

276.2.2

Several modules

(This section need to be updated. For now, see § Properties for the more general discussion.) It is possible to extend the definition to a tensor product of any number of modules over the same commutative ring. For example, the universal property of M1 ⊗ M2 ⊗ M3 is that each trilinear map on M1 × M2 × M3 → Z corresponds to a unique linear map M 1 ⊗ M 2 ⊗ M 3 → Z. The binary tensor product is associative: (M 1 ⊗ M 2 ) ⊗ M 3 is naturally isomorphic to M 1 ⊗ (M 2 ⊗ M 3 ). The tensor product of three modules defined by the universal property of trilinear maps is isomorphic to both of these iterated tensor products.

276.3 Properties Let R1 , R2 , R3 , R be rings, not necessarily commutative. • For a R1 -R2 -bimodule M 12 and a left R2 -module M 20 the tensor product

276.3. PROPERTIES

M12 ⊗R2 M20 is a left R1 -module. • For a right R2 -module M 02 and an R2 -R3 -bimodule M 23 the tensor product

M02 ⊗R2 M23 is a right R3 -module. • (associativity) For a right R1 -module M 01 , an R1 -R2 -bimodule M 12 , and a left R2 -module M 20 we have (M01 ⊗R1 M12 ) ⊗R2 M20 = M01 ⊗R1 (M12 ⊗R2 M20 ) .[5] • Since R is an R-R-bimodule, we have the tensor product

R ⊗R R = R with the ring multiplication mn =: m ⊗R n as its canonical balanced product. Let R be a commutative ring, and M, N and P be R-modules. Then • (identity)

R ⊗R M = M • (associativity) (M ⊗R N ) ⊗R P = M ⊗R (N ⊗R P ) [6] Thus M ⊗R N ⊗R P := M ⊗R (N ⊗R P ) is well-defined. • (symmetry)

M ⊗R N = N ⊗R M In fact, for any permutation σ of the set { 1, 2, …, n }, there is a unique isomorphism M1 ⊗R · · · ⊗R Mn → Mσ(1) ⊗R · · · ⊗R Mσ(n) under which x1 ⊗ · · · ⊗ xn maps to xσ(1) ⊗ · · · ⊗ xσ(n) .

825

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CHAPTER 276. TENSOR PRODUCT OF MODULES

• (distributive property)

M ⊗R (N ⊕ P ) = (M ⊗R N ) ⊕ (M ⊗R P ) In fact,

⊕ ⊕ M ⊗R ( Ni ) = M ⊗R Ni i∈I

i∈I

for an index set I of arbitrary cardinality. • (commutes with finite product) for any finitely many Ni ,

M ⊗R

n ∏

Ni =

i=1

n ∏

M ⊗R N i

i=1

• (commutes with localization) for any multiplicatively closed subset S of R,

S −1 (M ⊗R N ) = S −1 M ⊗S −1 R S −1 N as S −1 R -module. Since S −1 R is an R-algebra and S −1 − = S −1 R ⊗R − , this is a special case of: • (commutes with base extension) If S is an R-algebra, writing −S = S ⊗R − , (M ⊗R N )S = MS ⊗S NS ; [7] cf. § Extension of scalars. • (commutes with direct limit) for any direct system of R-modules Mi,

(lim Mi ) ⊗R N = lim(Mi ⊗R N ). −→ −→ f

g

• (tensoring is right exact) if 0 → N ′ →N →N ′′ → 0 is an exact sequence of R-modules, then

1⊗f

1⊗g

M ⊗R N ′ → M ⊗R N → M ⊗R N ′′ → 0 is an exact sequence of R-modules, where (1 ⊗ f )(x ⊗ y) = x ⊗ f (y). This is a consequence of: • (adjoint relation) HomR (M ⊗R N, P ) = HomR (M, HomR (N, P )) . • (tensor-hom relation) there is a canonical R-linear map:

HomR (M, N ) ⊗ P → HomR (M, N ⊗ P ), which is an isomorphism if either M or P is a finitely generated projective module (see § As linearitypreserving maps for the non-commutative case);[8] more generally, there is a canonical R-linear map: HomR (M, N ) ⊗ HomR (M ′ , N ′ ) → HomR (M ⊗ M ′ , N ⊗ N ′ ) which is an isomorphism if either (M, N ) or (M, M ′ ) is a pair of finitely generated projective modules.

276.3. PROPERTIES

827

To give a practical example, suppose M, N are free modules with bases ei , i ∈ I and fj , j ∈ J . Then M is the direct ⊕ sum M = i∈I Rei and the same for N. By the distributive property, one has:

M ⊗R N =

⊕

R(ei ⊗ fj )

i,j

i.e., ei ⊗ fj , i ∈ I, j ∈ J are the R-basis of M ⊗R N . Even if M is not free, a free presentation of M can be used to compute tensor products. The tensor product, in general, does not commute with inverse limit: on the one hand,

Q ⊗Z Z/pn = 0 (cf. “examples”). On the other hand,

(lim Z/pn ) ⊗Z Q = Zp ⊗Z Q = Zp [p−1 ] = Qp ←− where Zp , Qp are the ring of p-adic integers and the field of p-adic numbers. See also "profinite integer" for an example in the similar spirit. If R is not commutative, the order of tensor products could matter in the following way: we “use up” the right action of M and the left action of N to form the tensor product M ⊗R N ; in particular, N ⊗R M would not even be defined. If M, N are bi-modules, then M ⊗R N has the left action coming from the left action of M and the right action coming from the right action of N; those actions need not be the same as the left and right actions of N ⊗R M . The associativity holds more generally for non-commutative rings: if M is a right R-module, N a (R, S)-module and P a left S-module, then

(M ⊗R N ) ⊗S P = M ⊗R (N ⊗S P ) as abelian group. The general form of adjoint relation of tensor products says: if R is not necessarily commutative, M is a right Rmodule, N is a (R, S)-module, P is a right S-module, then as abelian group HomS (M ⊗R N, P ) = HomR (M, HomS (N, P )), f 7→ f ′ [9] where f ′ is given by f ′ (x)(y) = f (x ⊗ y). See also: tensor-hom adjunction.

276.3.1

Extension of scalars

Main article: extension of scalars The adjoint relation in the general form has an important special case: for any R-algebra S, M a right R-module, P a right S-module, using HomS (S, −) = − , we have the natural isomorphism:

HomS (M ⊗R S, P ) = HomR (M, ResR (P )). This says that the functor − ⊗R S is a left adjoint to the forgetful functor ResR , which restricts an S-action to an R-action. Because of this, − ⊗R S is often called the extension of scalars from R to S. In the representation theory, when R, S are group algebras, the above relation becomes the Frobenius reciprocity. Example: Rn ⊗R S = S n for any R-algebra S (i.e., a free module remains free after extending scalars.) Example: For a commutative ring R and a commutative R-algebra S, we have:

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CHAPTER 276. TENSOR PRODUCT OF MODULES

• S ⊗R R[x1 , . . . , xn ] = S[x1 , . . . , xn ] ; in fact, more generally, • S ⊗R (R[x1 , . . . , xn ]/I) = S[x1 , . . . , xn ]/IS[x1 , . . . , xn ], I an ideal. Example: Using the fact C = R[x]/(x2 + 1) , by the previous example and the Chinese remainder theorem, we have: as ring

C ⊗R C = C[x]/(x2 + 1) = C[x]/(x + i) × C[x]/(x − i) = C2 (this gives an example when a tensor product is a direct product.) Example: R ⊗Z Z[i] = C[i] = C where i is the imaginary unit. See also: Weil restriction.

276.4 Examples Let G be an abelian group in which every element has finite order (that is G is a torsion abelian group; for example G can be a finite abelian group or Q/Z). Then Q ⊗Z G = 0 .[10] Indeed, any element x of Q ⊗Z G is of the form

x=

∑

ri ⊗ gi

i

where ri ∈ Q, gi ∈ G . If ni is the order of gi , then we compute:

x=

∑ ∑ (ri /ni )ni ⊗ gi = ri /ni ⊗ ni gi = 0.

Similarly, one sees

Q/Z ⊗Z Q/Z = 0. Here are some useful identities: Let R be a commutative ring, I, J ideals, M, N R-modules. Then 1. R/I ⊗R M = M /IM . If M is flat, IM = I ⊗R M . 2. M /IM ⊗R/I N /IN = M ⊗R N ⊗R R/I. 3. R/I ⊗R R/J = R/(I + J) . Proof: Tensoring with M the exact sequence 0 → I → R → R/I → 0 gives

f

I ⊗R M →R ⊗R M = M → R/I ⊗R M → 0 where f is given by i ⊗ x 7→ ix . Since the image of f is IM, we get the first part of 1. If M is flat, f is injective and so is an isomorphism onto its image. 2. follows from 1. and 3. is because R/I ⊗R R/J =

R/J I(R/J)

=

R/J (I+J)/J

= R/(I + J) . □

276.5. CONSTRUCTION

829

Example: If G is an abelian group, G ⊗Z Z/n = G/nG ; this follows from 1. Example: Z/n ⊗Z Z/m = Z/gcd(n, m) ; this follows from 3. Example: Let µn be the group of n-th roots of unity. It is a cyclic group and cyclic groups are classified by orders. Thus, non-canonically, µn ≈ Z/n and thus, when g is the gcd of n and m,

µn ⊗Z µm ≈ µg . Example: Consider Q ⊗Z Q . Since Q ⊗Q Q is obtained from Q ⊗Z Q by imposing Q -linearity on the middle, we have the surjection

Q ⊗Z Q → Q ⊗Q Q whose kernel is generated by elements of the form rs x ⊗ y − x ⊗ rs y where r, s, x, u are integers and s is nonzero. Since r r s r x ⊗ y = x ⊗ y = x ⊗ y, s s s s the kernel actually vanishes; hence, Q ⊗Z Q = Q ⊗Q Q = Q. Example: We propose to compare R ⊗Z R and R ⊗R R . Like in the previous example, we have: R ⊗Z R = R ⊗Q R as abelian group and thus as Q-vector space (any Z-linear map between Q-vector spaces is Q-linear). As Q-vector space, R has dimension (cardinarity of a basis) of continuum. Hence, R ⊗Q R has a Q-basis indexed by a product of continuums; thus its Q-dimension is continuum. Hence, for dimension reason, there is a non-canonical isomorphism of Q-vector spaces:

R ⊗Z R ≈ R ⊗R R

276.5 Construction The construction of M ⊗ N takes a quotient of a free abelian group with basis the symbols m ∗ n, used here to denote the ordered pair (m, n), for m in M and n in N by the subgroup generated by all elements of the form 1. −m ∗ (n + n′) + m ∗ n + m ∗ n′ 2. −(m + m′) ∗ n + m ∗ n + m′ ∗ n 3. (m · r) ∗ n − m ∗ (r · n) where m, m′ in M, n, n′ in N, and r in R. The quotient map which takes m ∗ n = (m, n) to the coset containing m ∗ n; that is,

⊗ : M × N → M ⊗R N, (m, n) 7→ [m ∗ n] is balanced, and the subgroup has been chosen minimally so that this map is balanced. The universal property of ⊗ follows from the universal properties of a free abelian group and a quotient. More category-theoretically, let σ be the given right action of R on M; i.e., σ(m, r) = m · r and τ the left action of R of N. Then the tensor product of M and N over R can be defined as the coequalizer:

M ×R×N

σ×1

→ →

1×τ

⊗

M × N →M ⊗R N,

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together with the requirements m ⊗ (n + n′ ) = m ⊗ n + m ⊗ n′ , (m + m′ ) ⊗ n = m ⊗ n + m′ ⊗ n. If S is a subring of a ring R, then M ⊗R N is the quotient group of M ⊗S N by the subgroup generated by xr ⊗S y − x ⊗S ry, r ∈ R, x ∈ M, y ∈ N , where x ⊗S y is the image of (x, y) under ⊗ : M × N → M ⊗S N. In particular, any tensor product of R-modules can be constructed, if so desired, as a quotient of a tensor product of abelian groups by imposing the R-balanced product property. In the construction of the tensor product over a commutative ring R, the R-module structure can be built in from the start by forming the quotient of a free R-module by the submodule generated by the elements given above for the general construction, augmented by the elements r ⋅ (m ∗ n) − m ∗ (r ⋅ n). Alternately, the general construction can be given a Z(R)-module structure by defining the scalar action by r ⋅ (m ⊗ n) = m ⊗ (r ⋅ n) when this is well-defined, which is precisely when r ∈ Z(R), the centre of R. The direct product of M and N is rarely isomorphic to the tensor product of M and N. When R is not commutative, then the tensor product requires that M and N be modules on opposite sides, while the direct product requires they be modules on the same side. In all cases the only function from M × N to G that is both linear and bilinear is the zero map.

276.6 As linear maps In the general case, not all the properties of a tensor product of vector spaces extend to modules. Yet, some useful properties of the tensor product, considered as module homomorphisms, remain.

276.6.1

Dual module

See also: Duality (mathematics) § Dual objects The dual module of a right R-module E, is defined as HomR(E, R) with the canonical left R-module structure, and is denoted E ∗ .[11] The canonical structure is the pointwise operations of addition and scalar multiplication. Thus, E ∗ is the set of all R-linear maps E → R (also called linear forms), with operations

(ϕ + ψ)(u) = ϕ(u) + ψ(u), (r · ϕ)(u) = r · ϕ(u),

ϕ, ψ ∈ E ∗ , u ∈ E

ϕ ∈ E ∗ , u ∈ E, r ∈ R,

The dual of a left R-module is defined analogously, with the same notation. There is always a canonical homomorphism E → E ∗∗ from E to its second dual. It is an isomorphism if E is a free module of finite rank. In general, E is called a reflexive module if the canonical homomorphism is an isomorphism.

276.6.2

Duality pairing

We denote the natural pairing of its dual E ∗ and a right R-module E, or of a left R-module F and its dual F ∗ as ⟨·, ·⟩ : E ∗ × E → R : (e′ , e) 7→ ⟨e′ , e⟩ = e′ (e) ⟨·, ·⟩ : F × F ∗ → R : (f, f ′ ) 7→ ⟨f, f ′ ⟩ = f ′ (f ). The pairing is left R-linear in its left argument, and right R-linear in its right argument:

⟨r · g, h · s⟩ = r · ⟨g, h⟩ · s,

r, s ∈ R.

276.7. EXAMPLE FROM DIFFERENTIAL GEOMETRY: TENSOR FIELD

276.6.3

831

An element as a (bi)linear map

In the general case, each element of the tensor product of modules gives rise to a left R-linear map, to a right Rlinear map, and to an R-bilinear form. Unlike the commutative case, in the general case the tensor product is not an R-module, and thus does not support scalar multiplication. • Given right R-module E and right R-module F, there is a canonical homomorphism θ : F ⊗R E ∗ → HomR(E, F) such that θ(f ⊗ e′) is the map e ↦ f ⋅ ⟨e′, e⟩.[12] • Given left R-module E and right R-module F, there is a canonical homomorphism θ : F ⊗R E → HomR(E ∗ , F) such that θ(f ⊗ e) is the map e′ ↦ f ⋅ ⟨e, e′⟩.[13] Both cases hold for general modules, and become isomorphisms if the modules E and F are restricted to being finitely generated projective modules (in particular free modules of finite ranks). Thus, an element of a tensor product of modules over a ring R maps canonically onto an R-linear map, though as with vector spaces, constraints apply to the modules for this to be equivalent to the full space of such linear maps. • Given right R-module E and left R-module F, there is a canonical homomorphism θ : F ∗ ⊗R E ∗ → LR(F × E, R) such that θ(f′ ⊗ e′) is the map (f, e) ↦ ⟨f, f′⟩ ⋅ ⟨e′, e⟩. Thus, an element of a tensor product ξ ∈ F ∗ ⊗R E ∗ may be thought of giving rise to or acting as an R-bilinear map F × E → R.

276.6.4

Trace

Let R be a commutative ring and E an R-module. Then there is a canonical R-linear map:

E ∗ ⊗R E → R induced by ϕ ⊗ x 7→ ϕ(x) ; it is the unique R-linear corresponding to the duality pairing. If E is a finitely generated projective R-module, then one can identify E ∗ ⊗R E = EndR (E) through the canonical homomorphism mentioned above and then the above is the trace map:

tr : EndR (E) → R. When R is a field, this is the usual trace of a linear transformation.

276.7 Example from differential geometry: tensor field The most prominent example of a tensor product of modules in differential geometry is the tensor product of the spaces of vector fields and differential forms. More precisely, if R is the (commutative) ring of smooth functions on a smooth manifold M, then one puts

Tpq = Γ(M, T M )⊗p ⊗R Γ(M, T ∗ M )⊗q where Γ means the space of sections and the superscript ⊗p means tensoring p times over R. By definition, an element of Tpq is a tensor field of type (p, q). As R-modules, Tqp is the dual module of Tpq . [14] To lighten the notation, put E = Γ(M, T M ) and so E ∗ = Γ(M, T ∗ M ) .[15] When p, q ≥ 1, for each (k, l) with 1 ≤ k ≤ p, 1 ≤ l ≤ q, there is an R-multilinear map: cl , . . . , Xp , ω1 , . . . , ωbl , . . . , ωq ) E p × E ∗ q → E p−1 × E ∗ q−1 , (X1 , . . . , Xp , ω1 , . . . , ωq ) 7→ ⟨Xk , ωl ⟩(X1 , . . . , X

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where E p means R-linear map:

∏p 1

E and the hat means a term is omitted. By the universal property, it corresponds to a unique

Clk : Tpq → Tp−1 q−1 . It is called the contraction of tensors in the index (k, l). Unwinding what the universal property says one sees:

cl · · · ⊗ Xp ⊗ ω1 ⊗ · · · ωbl · · · ⊗ ωq . Clk (X1 ⊗ · · · ⊗ Xp ⊗ ω1 ⊗ · · · ⊗ ωq ) = ⟨Xk , ωl ⟩X1 ⊗ · · · X Remark: The preceding discussion is standard in textbooks on differential geometry (e.g., Helgason). In a way, the sheaf-theoretic construction (i.e., the language of sheaf of modules) is more natural and increasingly more common; for that, see the section § Tensor product of sheaves of modules.

276.8 Relationship to flat modules In general, − ⊗R − : Mod--R × R--Mod → Ab is a bifunctor which accepts a right and a left R module pair as input, and assigns them to the tensor product in the category of abelian groups. By fixing a right R module M, a functor M ⊗R − : R--Mod → Ab arises, and symmetrically a left R module N could be fixed to create a functor − ⊗R N : Mod--R → Ab . Unlike the Hom bifunctor HomR (−, −) , the tensor functor is covariant in both inputs. It can be shown that M⊗- and -⊗N are always right exact functors, but not necessarily left exact. By definition, a module T is a flat module if T⊗- is an exact functor. If {mi}i∈I and {nj}j∈J are generating sets for M and N, respectively, then {mi⊗nj}i∈I,j∈J will be a generating set for M⊗N. Because the tensor functor M⊗R- sometimes fails to be left exact, this may not be a minimal generating set, even if the original generating sets are minimal. If M is a flat module, the functor M⊗R- is exact by the very definition of a flat module. If the tensor products are taken over a field F, we are in the case of vector spaces as above. Since all F modules are flat, the bifunctor -⊗R- is exact in both positions, and the two given generating sets are bases, then {mi ⊗ nj | i ∈ I, j ∈ J} indeed forms a basis for M ⊗F N. When the tensor products are taken over a field F so that -⊗- is exact in both positions, and the generating sets are bases of M and N, it is true that {mi ⊗ nj | i ∈ I, j ∈ J} indeed forms a basis for M⊗F N. See also: pure submodule.

276.9 Additional structure If S and T are commutative R-algebras, then S ⊗R T will be a commutative R-algebra as well, with the multiplication map defined by (m1 ⊗ m2 ) (n1 ⊗ n2 ) = (m1 n1 ⊗ m2 n2 ) and extended by linearity. In this setting, the tensor product become a fibered coproduct in the category of R-algebras. If M and N are both R-modules over a commutative ring, then their tensor product is again an R-module. If R is a ring, RM is a left R-module, and the commutator rs − sr of any two elements r and s of R is in the annihilator of M, then we can make M into a right R module by setting mr = rm. The action of R on M factors through an action of a quotient commutative ring. In this case the tensor product of M with itself over R is again an R-module. This is a very common technique in commutative algebra.

276.10. GENERALIZATION

833

276.10 Generalization 276.10.1

Tensor product of complexes of modules

If X, Y are complexes of R-modules (R a commutative ring), then their tensor product is the complex given by (X ⊗R Y )n =

∑ i+j=n

Xi ⊗R Yi . [16]

For example, if C is a chain complex of flat abelian groups and if G is an abelian group, then the homology group of C ⊗Z G is the homology group of C with coefficients in G (see also: universal coefficient theorem.)

276.10.2

Tensor product of sheaves of modules

Main article: Sheaf of modules In this setup, for example, one can define a tensor field on a smooth manifold M as a (global or local) section of the tensor product (called tensor bundle)

(T M )⊗p ⊗O (T ∗ M )⊗q where O is the sheaf of rings of smooth functions on M and the bundles T M, T ∗ M are viewed as locally free sheaves on M. See also: http://www.encyclopediaofmath.org/index.php/Tensor_bundle The exterior bundle on M is the subbundle of the tensor bundle consisting of all antisymmetric covariant tensors. Sections of the exterior bundle are differential forms on M. See also: Tensor product bundle. One important case when one forms a tensor product over a sheaf of non-commutative rings appears in theory of D-modules; this sheaf of rings over which tensor products are formed is the sheaf of differential operators.

276.11 See also • Tor functor • Tensor product of algebras • Tensor product of fields

276.12 Notes [1] Hazewinkel, et al. (2004), p. 95, Prop. 4.5.1 [2] Bourbaki, ch. II §3.1 [3] First, if R = Z, then the claimed identification is given by f 7→ f ′ with f ′ (x)(y) = f (x, y) . In general, HomZ (N, G) has the structure of a right R-module by (g · r)(y) = g(ry) . Thus, for any Z-bilinear map f, f′ is R-linear ⇔ f ′ (xr) = f ′ (x) · r ⇔ f (xr, y) = f (x, ry). [4] Bourbaki, ch. II §3.2. [5] Bourbaki, ch. II §3.8 [6] The first three properties (plus identities on morphisms) say that the category of R-modules, with R commutative, forms a symmetric monoidal category. [7] Proof: (using associativity in a general form) (M ⊗R N )S = (S ⊗S M ) ⊗R N = MS ⊗R N = MS ⊗S S ⊗R N = MS ⊗S NS

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[8] Bourbaki, ch. II §4.4 [9] Bourbaki, ch.II §4.1 Proposition 1 [10] Example 3.6 of http://www.math.uconn.edu/~{}kconrad/blurbs/linmultialg/tensorprod.pdf [11] Bourbaki, ch. II §2.3 [12] Bourbaki, ch. II §4.2 eq. (11) [13] Bourbaki, ch. II §4.2 eq. (15) [14] Helgason, Lemma 2.3' [15] This is actually the definition of differential one-forms, global sections of T ∗ M , in Helgason, but is equivalent to the usual definition that does not use module theory. [16] May ch. 12 §3

276.13 References • Bourbaki, Algebra • Helgason, Sigurdur (1978), Differential geometry, Lie groups and symmetric spaces, Academic Press, ISBN 0-12-338460-5 • Northcott, D.G. (1984), Multilinear Algebra, Cambridge University Press, ISBN 613-0-04808-4. • Hazewinkel, Michiel; Gubareni, Nadezhda Mikhaĭlovna; Gubareni, Nadiya; Kirichenko, Vladimir V. (2004), Algebras, rings and modules, Springer, ISBN 978-1-4020-2690-4. • Peter May (1999), A concise course in algebraic topology, University of Chicago Press.

Chapter 277

Tight closure In mathematics, in the area of commutative algebra, tight closure is an operation defined on ideals in positive characteristic. It was introduced by Melvin Hochster and Craig Huneke (1988, 1990). Let R be a commutative noetherian ring containing a field of characteristic p > 0 . Hence p is a prime number. Let I be an ideal of R . The tight closure of I , denoted by I ∗ , is another ideal of R containing I . The ideal I ∗ is defined as follows. z ∈ I ∗ if and only if there exists a c ∈ R , where c is not contained in any minimal prime ideal of R , e e such that cz p ∈ I [p ] for all e ≫ 0 . If R is reduced, then one can instead consider all e > 0 . e

Here I [p ] is used to denote the ideal of R generated by the pe 'th powers of elements of I , called the e th Frobenius power of I . An ideal is called tightly closed if I = I ∗ . A ring in which all ideals are tightly closed is called weakly F -regular (for Frobenius regular). A previous major open question in tight closure is whether the operation of tight closure commutes with localization, and so there is the additional notion of F -regular, which says that all ideals of the ring are still tightly closed in localizations of the ring. Brenner & Monsky (2010) found a counterexample to the localization property of tight closure. However, there is still an open question of whether every weakly F -regular ring is F -regular. That is, if every ideal in a ring is tightly closed, is it true that every ideal in every localization of that ring also tightly closed?

277.1 References • Brenner, Holger; Monsky, Paul (2010), “Tight closure does not commute with localization”, Annals of Mathematics. Second Series 171 (1): 571–588, doi:10.4007/annals.2010.171.571, ISSN 0003-486X, MR 2630050 • Hochster, Melvin; Huneke, Craig (1988), “Tightly closed ideals”, American Mathematical Society. Bulletin. New Series 18 (1): 45–48, doi:10.1090/S0273-0979-1988-15592-9, ISSN 0002-9904, MR 919658 • Hochster, Melvin; Huneke, Craig (1990), “Tight closure, invariant theory, and the Briançon–Skoda theorem”, Journal of the American Mathematical Society 3 (1): 31–116, doi:10.2307/1990984, ISSN 0894-0347, MR 1017784

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Chapter 278

Top (mathematics) In the context of a module M over a ring R, the top of M is the largest semisimple quotient module of M if it exists. For finite-dimensional k-algebras (k a field), if rad(M) denotes the intersection of all proper maximal submodules of M (the radical of the module), then the top of M is M/rad(M). In the case of local rings with maximal ideal P, the top of M is M/PM. In general if R is a semilocal ring (=semi-artinian ring), that is, if R/Rad(R) is an Artinian ring, where Rad(R) is the Jacobson radical of R, then M/rad(M) is a semisimple module and is the top of M. This includes the cases of local rings and finite dimensional algebras over fields.

278.1 See also • Projective cover • Radical of a module • Socle (mathematics)

278.2 References • David Eisenbud, Commutative algebra with a view toward Algebraic Geometry ISBN 0-387-94269-6

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Chapter 279

Topological ring In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps R × R → R, where R × R carries the product topology.

279.1 General comments The group of units R× of R is a topological group when endowed with the topology coming from the embedding of R× into the product R × R as (x,x−1 ). However if the unit group is endowed with the subspace topology as a subspace of R, it may not be a topological group, because inversion on R× need not be continuous with respect to the subspace topology. An example of this situation is the adele ring of a global field; its unit group, called the idele group, is not a topological group in the subspace topology. If inversion on R× is continuous in the subspace topology of R then these two topologies on R× are the same. If one does not require a ring to have a unit, then one has to add the requirement of continuity of the additive inverse, or equivalently, to define the topological ring as a ring which is a topological group (for +) in which multiplication is continuous, too.

279.2 Examples Topological rings occur in mathematical analysis, for examples as rings of continuous real-valued functions on some topological space (where the topology is given by pointwise convergence), or as rings of continuous linear operators on some normed vector space; all Banach algebras are topological rings. The rational, real, complex and p-adic numbers are also topological rings (even topological fields, see below) with their standard topologies. In the plane, split-complex numbers and dual numbers form alternative topological rings. See hypercomplex numbers for other low-dimensional examples. In algebra, the following construction is common: one starts with a commutative ring R containing an ideal I, and then considers the I-adic topology on R: a subset U of R is open if and only if for every x in U there exists a natural number n such that x + I n ⊆ U. This turns R into a topological ring. The I-adic topology is Hausdorff if and only if the intersection of all powers of I is the zero ideal (0). The p-adic topology on the integers is an example of an I-adic topology (with I = (p)).

279.3 Completion Main article: Completion (ring theory)

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Every topological ring is a topological group (with respect to addition) and hence a uniform space in a natural manner. One can thus ask whether a given topological ring R is complete. If it is not, then it can be completed: one can find an essentially unique complete topological ring S which contains R as a dense subring such that the given topology on R equals the subspace topology arising from S. The ring S can be constructed as a set of equivalence classes of Cauchy sequences in R. The rings of formal power series and the p-adic integers are most naturally defined as completions of certain topological rings carrying I-adic topologies.

279.4 Topological fields Some of the most important examples are also fields F. To have a topological field we should also specify that inversion is continuous, when restricted to F\{0}. See the article on local fields for some examples.

279.5 References • L. V. Kuzmin (2001), “Topological ring”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • D. B. Shakhmatov (2001), “Topological field”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Seth Warner: Topological Rings. North-Holland, July 1993, ISBN 0-444-89446-2 • Vladimir I. Arnautov, Sergei T. Glavatsky and Aleksandr V. Michalev: Introduction to the Theory of Topological Rings and Modules. Marcel Dekker Inc, February 1996, ISBN 0-8247-9323-4. • N. Bourbaki, Éléments de Mathématique. Topologie Générale. Hermann, Paris 1971, ch. III §6

Chapter 280

Torsion (algebra) For other notions of torsion, see Torsion. In abstract algebra, the term torsion refers to elements of finite order in groups and to elements of modules annihilated by regular elements of a ring.

280.1 Definition An element m of a module M over a ring R is called a torsion element of the module if there exists a regular element r of the ring (an element that is neither a left nor a right zero divisor) that annihilates m, i.e., r m = 0. In an integral domain (a commutative ring without zero divisors), every non-zero element is regular, so a torsion element of a module over an integral domain is one annihilated by a non-zero element of the integral domain. Some authors use this as the definition of a torsion element but this definition does not work well over more general rings. A module M over a ring R is called a torsion module if all its elements are torsion elements, and torsion-free if zero is the only torsion element. If the ring R is an integral domain then the set of all torsion elements forms a submodule of M, called the torsion submodule of M, sometimes denoted T(M). If R is not commutative, T(M) may or may not be a submodule. It is shown in (Lam 2007) that R is a right Ore ring if and only if T(M) is a submodule of M for all right R modules. Since right Noetherian domains are Ore, this covers the case when R is a right Noetherian domain (which might not be commutative). More generally, let M be a module over a ring R and S be a multiplicatively closed subset of R. An element m of M is called an S-torsion element if there exists an element s in S such that s annihilates m, i.e., s m = 0. In particular, one can take for S the set of regular elements of the ring R and recover the definition above. An element g of a group G is called a torsion element of the group if it has finite order, i.e., if there is a positive integer m such that gm = e, where e denotes the identity element of the group, and gm denotes the product of m copies of g. A group is called a torsion (or periodic) group if all its elements are torsion elements, and a torsion-free group if the only torsion element is the identity element. Any abelian group may be viewed as a module over the ring Z of integers, and in this case the two notions of torsion coincide.

280.2 Examples 1. Let M be a free module over any ring R. Then it follows immediately from the definitions that M is torsion-free (if the ring R is not a domain then torsion is considered with respect to the set S of non-zero divisors of R). In particular, any free abelian group is torsion-free and any vector space over a field K is torsion-free when viewed as the module over K. 2. By contrast with Example 1, any finite group (abelian or not) is periodic and finitely generated. Burnside’s problem asks whether, conversely, any finitely generated periodic group must be finite. (The answer is “no” in general, even if the period is fixed.) 839

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3. In the modular group, Γ obtained from the group SL(2,Z) of two by two integer matrices with unit determinant by factoring out its center, any nontrivial torsion element either has order two and is conjugate to the element S or has order three and is conjugate to the element ST. In this case, torsion elements do not form a subgroup, for example, S · ST = T, which has infinite order. 4. The abelian group Q/Z, consisting of the rational numbers (mod 1), is periodic, i.e. every element has finite order. Analogously, the module K(t)/K[t] over the ring R = K[t] of polynomials in one variable is pure torsion. Both these examples can be generalized as follows: if R is a commutative domain and Q is its field of fractions, then Q/R is a torsion R-module. 5. The torsion subgroup of (R/Z,+) is (Q/Z,+) while the groups (R,+),(Z,+) are torsion-free. The quotient of a torsion-free abelian group by a subgroup is torsion-free exactly when the subgroup is a pure subgroup. 6. Consider a linear operator L acting on a finite-dimensional vector space V. If we view V as an F[L]-module in the natural way, then (as a result of many things, either simply by finite-dimensionality or as a consequence of the Cayley–Hamilton theorem), V is a torsion F[L]-module.

280.3 Case of a principal ideal domain Suppose that R is a (commutative) principal ideal domain and M is a finitely-generated R-module. Then the structure theorem for finitely generated modules over a principal ideal domain gives a detailed description of the module M up to isomorphism. In particular, it claims that

M ≃ F ⊕ T (M ), where F is a free R-module of finite rank (depending only on M) and T(M) is the torsion submodule of M. As a corollary, any finitely-generated torsion-free module over R is free. This corollary does not hold for more general commutative domains, even for R = K[x,y], the ring of polynomials in two variables. For non-finitely generated modules, the above direct decomposition is not true. The torsion subgroup of an abelian group may not be a direct summand of it.

280.4 Torsion and localization Assume that R is a commutative domain and M is an R-module. Let Q be the quotient field of the ring R. Then one can consider the Q-module

MQ = M ⊗R Q, obtained from M by extension of scalars. Since Q is a field, a module over Q is a vector space, possibly, infinitedimensional. There is a canonical homomorphism of abelian groups from M to MQ, and the kernel of this homomorphism is precisely the torsion submodule T(M). More generally, if S is a multiplicatively closed subset of the ring R, then we may consider localization of the R-module M,

MS = M ⊗R RS , which is a module over the localization RS. There is a canonical map from M to MS, whose kernel is precisely the S-torsion submodule of M. Thus the torsion submodule of M can be interpreted as the set of the elements that 'vanish in the localization'. The same interpretation continues to hold in the non-commutative setting for rings satisfying the Ore condition, or more generally for any right denominator set S and right R-module M.

280.5. TORSION IN HOMOLOGICAL ALGEBRA

841

280.5 Torsion in homological algebra The concept of torsion plays an important role in homological algebra. If M and N are two modules over a commutative ring R (for example, two abelian groups, when R = Z), Tor functors yield a family of R-modules Tori(M,N). The S-torsion of an R-module M is canonically isomorphic to Tor1 (M, RS/R). The symbol Tor denoting the functors reflects this relation with the algebraic torsion. This same result holds for non-commutative rings as well as long as the set S is a right denominator set.

280.6 Abelian varieties The torsion elements of an abelian variety are torsion points or, in an older terminology, division points. On elliptic curves they may be computed in terms of division polynomials.

280.7 See also • Analytic torsion • Arithmetic dynamics • Flat module • Localization of a module • Rank of an abelian group • Ray–Singer torsion • Torsion-free abelian group • Universal coefficient theorem

280.8 References • Ernst Kunz, “Introduction to Commutative algebra and algebraic geometry”, Birkhauser 1985, ISBN 0-81763065-1 • Irving Kaplansky, “Infinite abelian groups”, University of Michigan, 1954. • Michiel Hazewinkel (2001), “Torsion submodule”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Lam, T. Y. (2007), Exercises in modules and rings, Problem Books in Mathematics, New York: Springer, pp. xviii+412, doi:10.1007/978-0-387-48899-8, ISBN 0-387-98850-5, MR 2278849

842

The 4-torsion subgroup of an elliptic curve over the complex numbers.

CHAPTER 280. TORSION (ALGEBRA)

Chapter 281

Torsion-free module Not to be confused with Torsionless module. In algebra, a torsion-free module is a module over a ring such that 0 is the only element annihilated by a regular element (non zero-divisor) of the ring. In integral domains the regular elements of the ring are its nonzero elements, so in this case a torsion-free module is one such that 0 is the only element annihilated by some non-zero element of the ring. Some authors work only over integral domains and use this condition as the definition of a torsion-free module, but this does not work well over more general rings, for if the ring contains zero divisors then the only module satisfying this condition is the zero module.

281.1 Examples of torsion-free modules Over a commutative ring R with total quotient ring K, a module M is torsion-free if and only if Tor1 (K/R,M) vanishes. Therefore flat modules, and in particular free and projective modules, are torsion-free but the converse need not be true. An example of a torsion-free module that is not flat is the ideal (x,y) of the polynomial ring k[x,y] over a field k. • Any torsionless module is a torsion-free module, but the converse is not true, as Q is a torsion-free Z-module which is not torsionless.

281.2 Structure of torsion-free modules Over a Noetherian integral domain, torsion-free modules are the modules whose only associated prime is 0. More generally, over a Noetherian commutative ring the torsion-free modules are those all of whose associated primes are contained in associated primes of the ring. Over a Noetherian integrally closed domain, any finitely-generated torsion-free module has a free submodule such that the quotient by it is isomorphic to an ideal of the ring. Over a Dedekind domain, a finitely-generated module is torsion-free if and only if it is projective, but is in general not free. Any such module is isomorphic to the sum of a finitely-generated free module and an ideal, and the class of the ideal is uniquely determined by the module. Over a principal ideal domain, finitely-generated modules are torsion-free if and only if they are free.

281.3 See also • Torsion (algebra) • torsion-free abelian group 843

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• torsion-free abelian group of rank 1; the classification theory exists for this class.

281.4 References • Hazewinkel, Michiel, ed. (2001), “Torsion-free_module”, Encyclopedia of Mathematics, Springer, ISBN 9781-55608-010-4 • Matlis, Eben (1972), Torsion-free modules, The University of Chicago Press, Chicago-London, MR 0344237

Chapter 282

Torsionless module Not to be confused with Torsion-free module. In abstract algebra, a module M over a ring R is called torsionless if it can be embedded into some direct product RI . Equivalently, M is torsionless if each non-zero element of M has non-zero image under some R-linear functional f: f ∈ M ∗ = HomR (M, R),

f (m) ̸= 0.

This notion was introduced by Hyman Bass.

282.1 Properties and examples A module is torsionless if and only if the canonical map into its double dual, M → M ∗∗ = HomR (M ∗ , R),

m 7→ (f 7→ f (m)), m ∈ M, f ∈ M ∗ ,

is injective. If this map is bijective then the module is called reflexive. For this reason, torsionless modules are also known as semi-reflexive. • A free module is torsionless. More generally, a direct sum of torsionless modules is torsionless. • A free module is reflexive if it is finitely generated, but for some rings there are also infinitely generated free modules that are reflexive. For instance, the direct sum of countably many copies of the integers is a reflexive module over the integers, see for instance.[1] • A submodule of a torsionless module is torsionless. In particular, any projective module over R is torsionless; any left ideal of R is a torsionless left module, and similarly for the right ideals. • Any torsionless module over a domain is a torsion-free module, but the converse is not true, as Q is a torsionfree Z-module which is not torsionless. • If R is a commutative ring which is an integral domain and M is a finitely generated torsion-free module then M can be embedded into Rn and hence M is torsionless. • Suppose that N is a right R-module, then its dual N * has a structure of a left R-module. It turns out that any left R-module arising in this way is torsionless (similarly, any right R-module that is a dual of a left R-module is torsionless). • Over a Dedekind domain, a finitely generated module is reflexive if and only if it is torsion-free.[2] • Let R be a Noetherian ring and M a reflexive finitely generated module over R. Then M ⊗R S is a reflexive module over S whenever S is flat over R.[3] 845

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282.2 Relation with semihereditary rings Stephen Chase proved the following characterization of semihereditary rings in connection with torsionless modules: For any ring R, the following conditions are equivalent:[4] • R is left semihereditary. • All torsionless right R-modules are flat. • The ring R is left coherent and satisfies any of the four conditions that are known to be equivalent: • All right ideals of R are flat. • All left ideals of R are flat. • Submodules of all right flat R-modules are flat. • Submodules of all left flat R-modules are flat. (The mixture of left/right adjectives in the statement is not a mistake.)

282.3 See also • Prüfer domain • reflexive sheaf

282.4 References [1] P. C. Eklof and A. H. Mekler, Almost free modules, North-Holland Mathematical Library vol. 46, North-Holland, Amsterdam 1990 [2] Proof: If M is reflexive, it is torsionless, thus is a submodule of a finitely generated projective module and hence is projective (semi-hereditary condition). Conversely, over a Dedekind domain, a finitely generated torsion-free module is projective and a projective module is reflexive (the existence of a dual basis). [3] Bourbaki Ch. VII, § 4, n. 2. Proposition 8. [4] Lam 1999, p 146.

• Chapter VII of Bourbaki, Nicolas (1998), Commutative algebra (2nd ed.), Springer Verlag, ISBN 3-540-642390 • Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294

Chapter 283

Total ring of fractions In abstract algebra, the total quotient ring,[1] or total ring of fractions,[2] is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings R that may have zero divisors. The construction embeds R in a larger ring, giving every non-zero-divisor of R an inverse in the larger ring. Nothing more in A can be given an inverse, if one wants the homomorphism from A to the new ring to be injective.

283.1 Definition Let R be a commutative ring and let S be the set of elements which are not zero divisors in R ; then S is a multiplicatively closed set. Hence we may localize the ring R at the set S to obtain the total quotient ring S −1 R = Q(R) . If R is a domain, then S = R − {0} and the total quotient ring is the same as the field of fractions. This justifies the notation Q(R) , which is sometimes used for the field of fractions as well, since there is no ambiguity in the case of a domain. Since S in the construction contains no zero divisors, the natural map R → Q(R) is injective, so the total quotient ring is an extension of R .

283.2 Examples The total quotient ring Q(A×B) of a product ring is the product of total quotient rings Q(A)×Q(B) . In particular, if A and B are integral domains, it is the product of quotient fields. The total quotient ring of the ring of holomorphic functions on an open set D of complex numbers is the ring of meromorphic functions on D, even if D is not connected. In an Artinian ring, all elements are units or zero divisors. Hence the set of non-zero divisors is the group of units of the ring, R× , and so Q(R) = (R× )−1 R . But since all these elements already have inverses, Q(R) = R . The same thing happens in a commutative von Neumann regular ring R. Suppose a in R is not a zero divisor. Then in a von Neumann regular ring a = axa for some x in R, giving the equation a(xa − 1) = 0. Since a is not a zero divisor, xa = 1, showing a is a unit. Here again, Q(R) = R . • In algebraic geometry one considers a sheaf of total quotient rings on a scheme, and this may be used to give one possible definition of a Cartier divisor.

283.3 The total ring of fractions of a reduced ring There is an important fact: Proposition — Let A be a Noetherian reduced ring with the minimal prime ideals p1 , . . . , pr . Then 847

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Q(A) ≃

CHAPTER 283. TOTAL RING OF FRACTIONS

r ∏

Q(A/pi ).

1

Proof: Apply the Chinese remainder theorem to Q(A) with the maximal ideals pi Q(A) (note: pi Q(A) is maximal since if x in A is not in pi , then it is a nonzerodivisor and so is a unit in Q(A).) □

283.4 Generalization If R is a commutative ring and S is any multiplicative subset in R , the localization S −1 R can still be constructed, but the ring homomorphism from R to S −1 R might fail to be injective. For example, if 0 ∈ S , then S −1 R is the trivial ring.

283.5 Notes [1] Matsumura (1980), p. 12 [2] Matsumura (1989), p. 21

283.6 References • Hideyuki Matsumura, Commutative algebra, 1980 • Hideyuki Matsumura, Commutative ring theory, 1989

Chapter 284

Triangular matrix ring In algebra, a triangular matrix ring is a ring constructed from two rings and a bimodule.

284.1 Definition If T and U are rings and M is a U-T-bimodule, then the triangular matrix ring (T 0 M U) consists of 2 by 2 matrices (t 0 m u) with t ∈ T, m ∈ M, u ∈ U, with ordinary matrix multiplication and division.

284.2 References • Auslander, Maurice; Reiten, Idun; Smalø, Sverre O. (1997) [1995], Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, ISBN 978-0-521-59923-8, MR 1314422

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Uniform module In abstract algebra, a module is called a uniform module if the intersection of any two nonzero submodules is nonzero. This is equivalent to saying that every nonzero submodule of M is an essential submodule. A ring may be called a right (left) uniform ring if it is uniform as a right (left) module over itself. Alfred Goldie used the notion of uniform modules to construct a measure of dimension for modules, now known as the uniform dimension (or Goldie dimension) of a module. Uniform dimension generalizes some, but not all, aspects of the notion of the dimension of a vector space. Finite uniform dimension was a key assumption for several theorems by Goldie, including Goldie’s theorem, which characterizes which rings are right orders in a semisimple ring. Modules of finite uniform dimension generalize both Artinian modules and Noetherian modules. In the literature, uniform dimension is also referred to as simply the dimension of a module or the rank of a module. Uniform dimension should not be confused with the related notion, also due to Goldie, of the reduced rank of a module.

285.1 Properties and examples of uniform modules Being a uniform module is not usually preserved by direct products or quotient modules. The direct sum of two nonzero uniform modules always contains two submodules with intersection zero, namely the two original summand modules. If N 1 and N 2 are proper submodules of a uniform module M and neither submodule contains the other, then M /(N1 ∩ N2 ) fails to be uniform, as

N1 /(N1 ∩ N2 ) ∩ N2 /(N1 ∩ N2 ) = {0}. Uniserial modules are uniform, and uniform modules are necessarily directly indecomposable. Any commutative domain is a uniform ring, since if a and b are nonzero elements of two ideals, then the product ab is a nonzero element in the intersection of the ideals.

285.2 Uniform dimension of a module The following theorem makes it possible to define a dimension on modules using uniform submodules. It is a module version of a vector space theorem: Theorem: If Uᵢ and V are members of a finite collection of uniform submodules of a module M such that ⊕ni=1 Ui and ⊕m i=1 Vi are both essential submodules of M, then n = m. The uniform dimension of a module M, denoted u.dim(M), is defined to be n if there exists a finite set of uniform submodules Uᵢ such that ⊕ni=1 Ui is an essential submodule of M. The preceding theorem ensures that this n is well defined. If no such finite set of submodules exists, then u.dim(M) is defined to be ∞. When speaking of the uniform dimension of a ring, it is necessary to specify whether u.dim(RR) or rather u.dim(RR) is being measured. It is possible to have two different uniform dimensions on the opposite sides of a ring. 850

285.3. HOLLOW MODULES AND CO-UNIFORM DIMENSION

851

If N is a submodule of M, then u.dim(N) ≤ u.dim(M) with equality exactly when N is an essential submodule of M. In particular, M and its injective hull E(M) always have the same uniform dimension. It is also true that u.dim(M) = n if and only if E(M) is a direct sum of n indecomposable injective modules. It can be shown that u.dim(M) = ∞ if and only if M contains an infinite direct sum of nonzero submodules. Thus if M is either Noetherian or Artinian, M has finite uniform dimension. If M has finite composition length k, then u.dim(M) ≤ k with equality exactly when M is a semisimple module. (Lam 1999) A standard result is that a right Noetherian domain is a right Ore domain. In fact, we can recover this result from another theorem attributed to Goldie, which states that the following three conditions are equivalent for a domain D: • D is right Ore • u.dim(DD) = 1 • u.dim(DD) < ∞

285.3 Hollow modules and co-uniform dimension The dual notion of a uniform module is that of a hollow module: a module M is said to be hollow if, when N 1 and N 2 are submodules of M such that N1 + N2 = M , then either N 1 = M or N 2 = M. Equivalently, one could also say that every proper submodule of M is a superfluous submodule. These modules also admit an analogue of uniform dimension, called co-uniform dimension, corank, hollow dimension or dual Goldie dimension. Studies of hollow modules and co-uniform dimension were conducted in (Fleury 1974), (Reiter 1981), (Takeuchi 1976), (Varadarajan 1979) and (Miyashita 1966). The reader is cautioned that Fleury explored distinct ways of dualizing Goldie dimension. Varadarajan, Takeuchi and Reiter’s versions of hollow dimension are arguably the more natural ones. Grzeszczuk and Puczylowski in (Grzeszczuk & Puczylowski 1984) gave a definition of uniform dimension for modular lattices such that the hollow dimension of a module was the uniform dimension of its dual lattice of submodules. It is always the case that a finitely cogenerated module has finite uniform dimension. This raises the question: does a finitely generated module have finite hollow dimension? The answer turns out to be no: it was shown in (Sarath & Varadarajan 1979) that if a module M has finite hollow dimension, then M/J(M) is a semisimple, Artinian module. There are many rings with unity for which R/J(R) is not semisimple Artinian, and given such a ring R, R itself is finitely generated but has infinite hollow dimension. Sarath and Varadarajan showed later, that M/J(M) being semisimple Artinian is also sufficient for M to have finite hollow dimension provided J(M) is a superfluous submodule of M.[1] This shows that the rings R with finite hollow dimension either as a left or right R-module are precisely the semilocal rings. An additional corollary of Varadarajan’s result is that RR has finite hollow dimension exactly when RR does. This contrasts the finite uniform dimension case, since it is known a ring can have finite uniform dimension on one side and infinite uniform dimension on the other.

285.4 Textbooks • Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294

285.5 Primary sources [1] The same result can be found in (Reiter 1981) and (Hanna & Shamsuddin 1984)

• Fleury, Patrick (1974), “A note on dualizing Goldie dimension”, Canadian Mathematical Bulletin 17: 511–517, doi:10.4153/cmb-1974-090-0 • Goldie, A. W. (1958), “The structure of prime rings under ascending chain conditions”, Proc. London Math. Soc. (3) 8: 589–608, ISSN 0024-6115, MR 0103206, (21 \#1988)

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• Goldie, A. W. (1960), “Semi-prime rings with maximum condition”, Proc. London Math. Soc. (3) 10: 201– 220, ISSN 0024-6115, MR MR0111766, (22 \#2627) • Grezeszcuk, P; Puczylowski,E (1984), “On Goldie and dual Goldie dimension,”, Journal of Pure and Applied Algebra 31: 47–55, doi:10.1016/0022-4049(84)90075-6 • Hanna, A.; Shamsuddin, A. (1984), “Duality in the category of modules. Applications,”, Algebra Berichte 49 (Verlag Reinhard Fischer, Munchen) • Miyashita, Y. (1966), “Quasi-projective modules, perfect modules, and a theorem for modular lattices”, J. Fac. Sci. Hokkaido Ser. I (contd. as Hokkaido Journal of Mathematics) 19: 86–110, MR 0213390, (35 \#4254) • Reiter, E. (1981), “A dual to the Goldie ascending chain condition on direct sums of submodules”, Bull. Calcutta Math. Soc. 73: 55–63 • Sarath B.; Varadarajan, K. (1979), “Dual Goldie dimension II”, Communications in Algebra 7 (17): 1885–1899, doi:10.1080/00927877908822434 • Takeuchi, T. (1976), “On cofinite-dimensional modules.”, Hokkaido Journal of Mathematics 5 (1): 1–43, doi:10.14492/hokmj/1381758746, ISSN 0385-4035, MR 0213390, (35 \#4254) • Varadarajan, K. (1979), “Dual Goldie dimension”, Comm. Algebra 7 (6): 565–610, doi:10.1080/00927877908822364, ISSN 0092-7872, MR MR524269, (80d:16014)

Chapter 286

Unipotent This article is about the algebraic term. For a biological cell having the capacity to develop into only one cell type, see Cell potency#Unipotency. In mathematics, a unipotent element, r, of a ring, R, is one such that r − 1 is a nilpotent element; in other words, (r − 1)n is zero. In particular, a square matrix, M, is a unipotent matrix, if and only if its characteristic polynomial, P(t), is a power of t − 1. Equivalently, M is unipotent if all its eigenvalues are 1. The term quasi-unipotent means that some power is unipotent, for example for a diagonalizable matrix with eigenvalues that are all roots of unity. In an unipotent affine algebraic group all elements are unipotent (see below for the definition of an element being unipotent in such a group).

286.1 Unipotent algebraic groups An element, x, of an affine algebraic group is unipotent when its associated right translation operator, rx, on the affine coordinate ring A[G] of G is locally unipotent as an element of the ring of linear endomorphism of A[G]. (Locally unipotent means that its restriction to any finite-dimensional stable subspace of A[G] is unipotent in the usual ring sense.) An affine algebraic group is called unipotent if all its elements are unipotent. Any unipotent algebraic group is isomorphic to a closed subgroup of the group of upper triangular matrices with diagonal entries 1, and conversely any such subgroup is unipotent. In particular any unipotent group is a nilpotent group, though the converse is not true (counterexample: the diagonal matrices of GLn(k)). If a unipotent group acts on an affine variety, all its orbits are closed, and if it acts linearly on a finite-dimensional vector space then it has a non-zero fixed vector. In fact, the latter property characterizes unipotent groups. Unipotent groups over an algebraically closed field of any given dimension can in principle be classified, but in practice the complexity of the classification increases very rapidly with the dimension, so people tend to give up somewhere around dimension 6. Over the real numbers (or more generally any field of characteristic 0) the exponential map takes any nilpotent square matrix to a unipotent matrix. Moreover, if U is a commutative unipotent group, the exponential map induces an isomorphism from the Lie algebra of U to U itself.

286.2 Unipotent radical The unipotent radical of an algebraic group G is the set of unipotent elements in the radical of G. It is a connected unipotent normal subgroup of G, and contains all other such subgroups. A group is called reductive if its unipotent radical is trivial. If G is reductive then its radical is a torus. 853

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286.3 Jordan decomposition Main article: Jordan–Chevalley decomposition Any element g of a linear algebraic group over a perfect field can be written uniquely as the product g = gugs of commuting unipotent and semisimple elements gu and gs. In the case of the group GLn(C), this essentially says that any invertible complex matrix is conjugate to the product of a diagonal matrix and an upper triangular one, which is (more or less) the multiplicative version of the Jordan–Chevalley decomposition. There is also a version of the Jordan decomposition for groups: any commutative linear algebraic group over a perfect field is the product of a unipotent group and a semisimple group.

286.4 See also • unipotent representation • Deligne–Lusztig theory

286.5 References • A. Borel, Linear algebraic groups, ISBN 0-387-97370-2 • Borel, Armand (1956), “Groupes linéaires algébriques”, Annals of Mathematics. Second Series (Annals of Mathematics) 64 (1): 20–82, doi:10.2307/1969949, JSTOR 1969949 • Popov, V.L. (2001), “unipotent element”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Popov, V.L. (2001), “unipotent group”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Suprunenko, D.A. (2001), “unipotent matrix”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 287

Unique factorization domain “Unique factorization” redirects here. For the uniqueness of integer factorization, see fundamental theorem of arithmetic. In mathematics, a unique factorization domain (UFD) is a commutative ring in which every non-zero non-unit element can be written as a product of prime elements (or irreducible elements), uniquely up to order and units, analogous to the fundamental theorem of arithmetic for the integers. UFDs are sometimes called factorial rings, following the terminology of Bourbaki. Unique factorization domains appear in the following chain of class inclusions: Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields

287.1 Definition Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero element x of R can be written as a product (an empty product if x is a unit) of irreducible elements pᵢ of R and a unit u: x = u p1 p2 ... pn with n ≥ 0 and this representation is unique in the following sense: If q1 ,...,qm are irreducible elements of R and w is a unit such that x = w q1 q2 ... qm with m ≥ 0, then m = n, and there exists a bijective map φ : {1,...,n} → {1,...,m} such that pi is associated to qᵩ₍i₎ for i ∈ {1, ..., n}. The uniqueness part is usually hard to verify, which is why the following equivalent definition is useful: A unique factorization domain is an integral domain R in which every non-zero element can be written as a product of a unit and prime elements of R.

287.2 Examples Most rings familiar from elementary mathematics are UFDs: • All principal ideal domains, hence all Euclidean domains, are UFDs. In particular, the integers (also see fundamental theorem of arithmetic), the Gaussian integers and the Eisenstein integers are UFDs. 855

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• If R is a UFD, then so is R[X], the ring of polynomials with coefficients in R. Unless R is a field, R[X] is not a principal ideal domain. By iteration, a polynomial ring in any number of variables over any UFD (and in particular over a field) is a UFD. • The Auslander–Buchsbaum theorem states that every regular local ring is a UFD. • The formal power series ring K[[X1 ,...,Xn]] over a field K (or more generally over a regular UFD such as a PID) is a UFD. On the other hand, the formal power series ring over a UFD need not be a UFD, even if the UFD is local. For example, if R is the localization of k[x,y,z]/(x2 + y3 + z7 ) at the prime ideal (x,y,z) then R is a local ring that is a UFD, but the formal power series ring R[[X]] over R is not a UFD. • Mori showed that if the completion of a Zariski ring, such as a Noetherian local ring, is a UFD, then the ring is a UFD.[1] The converse of this is not true: there are Noetherian local rings that are UFDs but whose completions are not. The question of when this happens is rather subtle: for example, for the localization of k[x,y,z]/(x2 + y3 + z5 ) at the prime ideal (x,y,z), both the local ring and its completion are UFDs, but in the apparently similar example of the localization of k[x,y,z]/(x2 + y3 + z7 ) at the prime ideal (x,y,z) the local ring is a UFD but its completion is not. • Let R be any field of characteristic not 2. Klein and Nagata showed that the ring R[X1 ,...,Xn]/Q is a UFD whenever Q is a nonsingular quadratic form in the X's and n is at least 5. When n=4 the ring need not be a UFD. For example, R[X, Y, Z, W ]/(XY − ZW ) is not a UFD, because the element XY equals the element ZW so that XY and ZW are two different factorizations of the same element into irreducibles. • The ring of formal power series over the complex numbers is factorial, but the subring of those that converge everywhere, in other words the ring of entire functions in a single complex variable is not a UFD, since there exist entire functions with an infinity of zeros, and thus an infinity of irreducible factors, while a UFD factorization must be finite, e.g.: ) ∞ ( ∏ z2 sin πz = πz 1− 2 . n n=1 • The ring Q[x,y]/(x2 + 2y2 + 1) is factorial, but the ring Q(i)[x,y]/(x2 + 2y2 + 1) is not. On the other hand, The ring Q[x,y]/(x2 + y2 – 1) is not factorial, but the ring Q(i)[x,y]/(x2 + y2 – 1) is (Samuel 1964, p.35). Similarly the coordinate ring R[X,Y,Z]/(X2 + Y 2 + Z 2 − 1) of the 2-dimensional real sphere is factorial, but the coordinate ring C[X,Y,Z]/(X2 + Y 2 + Z 2 − 1) of the complex sphere is not. • Suppose that the variables Xi are given weights wi, and F(X1 ,...,Xn) is a homogeneous polynomial of weight w. Then if c is coprime to w and R is a UFD and either every finitely generated projective module over R is free or c is 1 mod w, the ring R[X1 ,...,Xn,Z]/(Z c − F(X1 ,...,Xn)) is a factorial ring (Samuel 1964, p.31). Non-example: √ √ • The quadratic integer ring Z[ −5] of all complex numbers b −5 ) ( a +√ ) , where a and b are integers, ( of the √ form is not a UFD because 6 factors as both (2)(3) and as 1 + −5 1 − −5 . These truly are different √ √ factorizations, because the only units in this ring are 1 and −1; thus, none of 2, 3, 1 + −5 , and 1 − −5 are associate. It is not hard to show that all four factors are irreducible as well, though this may not be obvious.[2] See also algebraic integer.

287.3 Properties Some concepts defined for integers can be generalized to UFDs: • In UFDs, every irreducible element is prime. (In any integral domain, every prime element is irreducible, but the converse does not always hold. For instance, the element z ∈ K[x, y, z]/(z 2 − xy) is irreducible, but not prime.) Note that this has a partial converse: a domain satisfying the ACCP is a UFD if and only if every irreducible element is prime.

287.4. EQUIVALENT CONDITIONS FOR A RING TO BE A UFD

857

• Any two (or finitely many) elements of a UFD have a greatest common divisor and a least common multiple. Here, a greatest common divisor of a and b is an element d which divides both a and b, and such that every other common divisor of a and b divides d. All greatest common divisors of a and b are associated. • Any UFD is integrally closed. In other words, if R is a UFD with quotient field K, and if an element k in K is a root of a monic polynomial with coefficients in R, then k is an element of R. • Let S be a multiplicatively closed subset of a UFD A. Then the localization S −1 A is a UFD. A partial converse to this also holds; see below.

287.4 Equivalent conditions for a ring to be a UFD A Noetherian integral domain is a UFD if and only if every height 1 prime ideal is principal (a proof is given below). Also, a Dedekind domain is a UFD if and only if its ideal class group is trivial. In this case it is in fact a principal ideal domain. There are also equivalent conditions for non-noetherian integral domains. Let A be an integral domain. Then the following are equivalent. 1. A is a UFD. 2. Every nonzero prime ideal of A contains a prime element. (Kaplansky) 3. A satisfies ascending chain condition on principal ideals (ACCP), and the localization S −1 A is a UFD, where S is a multiplicatively closed subset of A generated by prime elements. (Nagata criterion) 4. A satisfies (ACCP) and every irreducible is prime. 5. A is atomic and every irreducible is prime. 6. A is a GCD domain (i.e., any two elements have a greatest common divisor) satisfying (ACCP). 7. A is a Schreier domain,[3] and atomic. 8. A is a pre-Schreier domain and atomic. 9. A has a divisor theory in which every divisor is principal. 10. A is a Krull domain in which every divisorial ideal is principal (in fact, this is the definition of UFD in Bourbaki.) 11. A is a Krull domain and every prime ideal of height 1 is principal.[4] In practice, (2) and (3) are the most useful conditions to check. For example, it follows immediately from (2) that a PID is a UFD, since, in a PID, every prime ideal is generated by a prime element. For another example, consider a Noetherian integral domain in which every height one prime ideal is principal. Since every prime ideal has finite height, it contains height one prime ideal (induction on height), which is principal. By (2), the ring is a UFD.

287.5 See also • Parafactorial local ring

287.6 References [1] Bourbaki, 7.3, no 6, Proposition 4. [2] Artin, Michael (2011). Algebra. Prentice Hall. p. 360. ISBN 978-0-13-241377-0.

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[3] A Schreier domain is an integrally closed integral domain where, whenever x divides yz, x can be written as x = x1 x2 so that x1 divides y and x2 divides z. In particular, a GCD domain is a Schreier domain [4] Bourbaki, 7.3, no 2, Theorem 1.

• N. Bourbaki. Commutative algebra. • B. Hartley; T.O. Hawkes (1970). Rings, modules and linear algebra. Chapman and Hall. ISBN 0-412-09810-5. Chap. 4. • Chapter II.5 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-201-55540-0, Zbl 0848.13001 • David Sharpe (1987). Rings and factorization. Cambridge University Press. ISBN 0-521-33718-6. • Samuel, Pierre (1964), Murthy, M. Pavman, ed., Lectures on unique factorization domains, Tata Institute of Fundamental Research Lectures on Mathematics 30, Bombay: Tata Institute of Fundamental Research, MR 0214579 • Samuel, Pierre (1968). “Unique factorization”. The American Mathematical Monthly 75: 945–952. doi:10.2307/2315529. ISSN 0002-9890.

Chapter 288

Unit (ring theory) Not to be confused with Unit ring. In mathematics, an invertible element or a unit in a (unital) ring R is any element u that has an inverse element in the multiplicative monoid of R, i.e. an element v such that uv = vu = 1R, where 1R is the multiplicative identity.[1][2] The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation. It never contains the element 0 (except in the case of the zero ring), and is therefore not closed under addition; its complement however might be a group under addition, which happens if and only if the ring is a local ring. The term unit is also used to refer to the identity element 1R of the ring, in expressions like ring with a unit or unit ring, and also e.g. 'unit' matrix. For this reason, some authors call 1R “unity” or “identity”, and say that R is a “ring with unity” or a “ring with identity” rather than a “ring with a unit”. The multiplicative identity 1R and its opposite −1R are always units. Hence, pairs of additive inverse elements[3] x and −x are always associated.

288.1 Group of units Main article: Green’s relations § The H and D relations The units of R form a group U(R) under multiplication, the group of units of R. Other common notations for U(R) are R∗ , R× , and E(R) (for the German term Einheit). In a commutative unital ring R, the group of units U(R) acts on R via multiplication. The orbits of this action are called sets of associates; in other words, there is an equivalence relation ∼ on R called associatedness such that r∼s means that there is a unit u with r = us. One can check that U is a functor from the category of rings to the category of groups: every ring homomorphism f : R → S induces a group homomorphism U(f) : U(R) → U(S), since f maps units to units. This functor has a left adjoint which is the integral group ring construction. In an integral domain the cardinality of an equivalence class of associates is the same as that of U(R). A ring R is a division ring if and only if U(R) = R ∖ {0} . 859

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CHAPTER 288. UNIT (RING THEORY)

288.2 Examples • In the ring of integers Z, the only units are +1 and −1. • In the ring Z/nZ of integers modulo n, the units are the congruence classes (mod n) represented by integers coprime to n. They constitute the multiplicative group of integers modulo n. • Any root of unity in a ring R is a unit. (If rn = 1, then rn − 1 is a multiplicative inverse of r.) • If R is the ring of integers in a number field, Dirichlet’s unit theorem implies that the unit group of R is a finitely generated abelian group. For example, we have (√5 + 2)(√5 − 2) = 1 in the ring Z[1 + √5/2], and in fact the unit group of this ring is infinite. In general, the unit group of (the ring of integers of) a real quadratic field is infinite (of rank 1). • The unit group of the ring Mn(F) of n × n matrices over a field F is the group GLn(F) of invertible matrices.

288.3 References [1] Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9. [2] Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X. [3] In a ring, the additive inverse of a non-zero element can equal to the element itself.

Chapter 289

Universal enveloping algebra For the universal enveloping W* algebra of a C* algebra, see Sherman–Takeda theorem. In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U(L). This construction passes from the non-associative structure L to a (more familiar, and possibly easier to handle) unital associative algebra which captures the important properties of L. Any associative algebra A over the field K becomes a Lie algebra over K with the Lie bracket:

[a, b] = ab − ba That is, from an associative product, one can construct a Lie bracket by taking the commutator with respect to that associative product. Denote this Lie algebra by AL. Construction of the universal enveloping algebra attempts to reverse this process: to a given Lie algebra L over K, find the “most general” unital associative K-algebra A such that the Lie algebra AL contains L; this algebra A is U(L). The important constraint is to preserve the representation theory: the representations of L correspond in a one-to-one manner to the modules over U(L). In a typical context where L is acting by infinitesimal transformations, the elements of U(L) act like differential operators, of all orders. Next to Lie algebras, the construction of the universal enveloping algebra has been generalized for Malcev algebras,[1] Bol algebras [2] and left alternative algebras.[3]

289.1 Motivation An important topic in Lie algebras studies and probably the main source of their appearance in applications is representation of the Lie algebra. A representation ρ assigns to any element x of a Lie algebra a linear operator ρ(x). The space of linear operators is not only a Lie algebra, but also an associative algebra and so one can consider products ρ(x)ρ(y). The main point to introduce the universal enveloping algebra is to study such products in various representations of a Lie algebra. One obstacle can be immediately seen in a naive attempt to do this: properties of products drastically depend on the representation, not only on the Lie algebra itself. For example for one representation we might have ρ(x)ρ(y) = 0, while in another representation this product may not be zero. Nevertheless it appears to be true that certain properties are universal for all representations, i.e. they hold true for all representations simultaneously. The universal enveloping algebra is a way to grasp all such properties and only them.

289.2 Universal property Let X be any Lie algebra over K. Given a unital associative K-algebra U and a Lie algebra homomorphism: h: X → UL, (notation as above) we say that U is the universal enveloping algebra of X if it satisfies the following universal property: for any unital associative K-algebra A and Lie algebra homomorphism f: X → AL there exists a unique unital algebra homomorphism g: U → A such that: f(-) = gL (h(-)). 861

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This is the universal property expressing that the functor sending X to its universal enveloping algebra is left adjoint to the functor sending a unital associative algebra A to its Lie algebra AL.

289.3 Direct construction From this universal property, one can prove that if a Lie algebra has a universal enveloping algebra, then this enveloping algebra is uniquely determined by L (up to a unique algebra isomorphism). By the following construction, which suggests itself on general grounds (for instance, as part of a pair of adjoint functors), we establish that indeed every Lie algebra does have a universal enveloping algebra. Starting with the tensor algebra T(L) on the vector space underlying L, we take U(L) to be the quotient of T(L) made by imposing the relations a ⊗ b − b ⊗ a = [a, b] for all a and b in (the image in T(L) of) L, where the bracket on the RHS means the given Lie algebra product, in L. Formally, we define

U (L) = T (L)/I where I is the two-sided ideal of T(L) generated by elements of the form a ⊗ b − b ⊗ a − [a, b],

a, b ∈ L.

The natural map L → T(L) descends to a map h : L → U(L), and this is the Lie algebra homomorphism used in the universal property given above. The analogous construction for Lie superalgebras is straightforward.

289.4 Examples in particular cases If L is abelian (that is, the bracket is always 0), then U(L) is commutative; if a basis of the vector space L has been chosen, then U(L) can be identified with the polynomial algebra over K, with one variable per basis element. If L is the Lie algebra corresponding to the Lie group G, U(L) can be identified with the algebra of left-invariant differential operators (of all orders) on G; with L lying inside it as the left-invariant vector fields as first-order differential operators. To relate the above two cases: if L is a vector space V as abelian Lie algebra, the left-invariant differential operators are the constant coefficient operators, which are indeed a polynomial algebra in the partial derivatives of first order. The center of U(L) is called Z(L) and consists of the left- and right- invariant differential operators; this in the case of G not commutative will often not be generated by first-order operators (see for example Casimir operator of a semi-simple Lie algebra). Another characterisation in Lie group theory is of U(L) as the convolution algebra of distributions supported only at the identity element e of G. The algebra of differential operators in n variables with polynomial coefficients may be obtained starting with the Lie algebra of the Heisenberg group. See Weyl algebra for this; one must take a quotient, so that the central elements of the Lie algebra act as prescribed scalars.

289.5 Further description of structure The fundamental Poincaré–Birkhoff–Witt theorem gives a precise description of U(L); the most important consequence is that L can be viewed as a linear subspace of U(L). More precisely: the canonical map h : L → U(L) is always injective. Furthermore, U(L) is generated as a unital associative algebra by L.

289.6. SEE ALSO

863

L acts on itself by the Lie algebra adjoint representation, and this action can be extended to a representation of L on U(L): L acts as an algebra of derivations on T(L), and this action respects the imposed relations, so it actually acts on U(L). (This is the purely infinitesimal way of looking at the invariant differential operators mentioned above.) Under this representation, the elements of U(L) invariant under the action of L (i.e. such that any element of L acting on them gives zero) are called invariant elements. They are generated by the Casimir invariants. As mentioned above, the construction of universal enveloping algebras is part of a pair of adjoint functors. U is a functor from the category of Lie algebras over K to the category of unital associative K-algebras. This functor is left adjoint to the functor which maps an algebra A to the Lie algebra AL. The universal enveloping algebra construction is not exactly inverse to the formation of AL: if we start with an associative algebra A, then U(AL) is not equal to A; it is much bigger. The facts about representation theory mentioned earlier can be made precise as follows: the abelian category of all representations of L is isomorphic to the abelian category of all left modules over U(L). The construction of the group algebra for a given group is in many ways analogous to constructing the universal enveloping algebra for a given Lie algebra. Both constructions are universal and translate representation theory into module theory. Furthermore, both group algebras and universal enveloping algebras carry natural comultiplications which turn them into Hopf algebras. The center of the universal enveloping algebra of a simple Lie algebra is described by the Harish-Chandra isomorphism.

289.6 See also • Poincaré–Birkhoff–Witt theorem • Harish-Chandra homomorphism

289.7 References [1] J.M. Perez-Izquierdo, I.P. Shestakov: An envelope for Malcev algebras, Journal of Algebra 272 (2004) 379–393. [2] J.M. Perez-Izquierdo: An envelope for Bol algebras, Journal of Algebra 284 (2005) 480–493. [3] Rukavicka Josef: An envelope for left alternative algebras, International Journal of Algebra, Vol. 7, 2013, no. 10, 455–462,

• Dixmier, Jacques (1996) [1974], Enveloping algebras, Graduate Studies in Mathematics 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0560-2, MR 0498740 • Musson, Ian M. (2012), Lie Superalgebras and Enveloping Algebras, Graduate Studies in Mathematics 131, Providence, R.I.: American Mathematical Society, ISBN 0-8218-6867-5, Zbl 1255.17001

Chapter 290

Universal geometric algebra In mathematics, a universal geometric algebra is a type of geometric algebra generated by real vector spaces endowed with an indefinite quadratic form. Some authors restrict this to the infinite-dimensional case. The universal geometric algebra G(n, n) of order 22n is defined as the Clifford algebra of 2n-dimensional pseudoEuclidean space Rn, n .[1] This algebra is also called the “mother algebra”. It has a nondegenerate signature. The vectors in this space generate the algebra through the geometric product. This product makes the manipulation of vectors more similar to the familiar algebraic rules, although non-commutative. When n = ∞, i.e. there are countably many dimensions, then G(∞, ∞) is called simply the universal geometric algebra (UGA), which contains vector spaces such as Rp, q and their respective geometric algebras G(p, q) . A special case is the algebra of spacetime, STA. UGA contains all finite-dimensional geometric algebras (GA). The elements of UGA are called multivectors. Every multivector can be written as the sum of several r-vectors. Some r-vectors are scalars (r = 0), vectors (r = 1) and bivectors (r = 2). Scalars are identical to the real numbers. Complex number are not used as scalars because there already exist structures in UGA that are equivalent to the complex numbers. One may generate a finite-dimensional GA by choosing a unit pseudoscalar (I). The set of all vectors that satisfy

a∧I =0 is a vector space. The geometric product of the vectors in this vector space then defines the GA, of which I is a member. Since every finite-dimensional GA has a unique I (up to a sign), one can define or characterize the GA by it. A pseudoscalar can be interpreted as an n-plane segment of unit area in an n-dimensional vector space.

290.1 Vector manifolds A vector manifold is a special set of vectors in the UGA.[2] These vectors generate a set of linear spaces tangent to the vector manifold. Vector manifolds were introduced to do calculus on manifolds so one can define (differentiable) manifolds as a set isomorphic to a vector manifold. The difference lies in that a vector manifold is algebraically rich while a manifold is not. Since this is the primary motivation for vector manifolds the following interpretation is rewarding. Consider a vector manifold as a special set of “points”. These points are members of an algebra and so can be added and multiplied. These points generate a tangent space of definite dimension “at” each point. This tangent space generates a (unit) pseudoscalar which is a function of the points of the vector manifold. A vector manifold is characterized by its pseudoscalar. The pseudoscalar can be interpreted as a tangent oriented n-plane segment of unit area. Bearing this in mind, a manifold looks locally like Rn at every point. Although a vector manifold can be treated as a completely abstract object, a geometric algebra is created so that every element of the algebra represents a geometric object and algebraic operations such as adding and multiplying correspond to geometric transformations. 864

290.2. SEE ALSO

865

Consider a set of vectors {x} = M n in UGA. If this set of vectors generates a set of “tangent” simple (n + 1)-vectors, which is to say

∀x ∈ M n : ∃In (x) = x ∧ A(x) | In (x) ∨ Mn = x then M n is a vector manifold, the value of A is that of a simple n-vector. If one interprets these vectors as points then In(x) is the pseudoscalar of an algebra tangent to M n at x. In(x) can be interpreted as a unit area at an oriented n-plane: this is why it is labeled with n. The function In gives a distribution of these tangent n-planes over M n . A vector manifold is defined similarly to how a particular GA can be defined, by its unit pseudoscalar. The set {x} is not closed under addition and multiplication by scalars. This set is not a vector space. At every point the vectors generate a tangent space of definite dimension. The vectors in this tangent space are different from the vectors of the vector manifold. In comparison to the original set they are bivectors, but since they span a linear space—the tangent space—they are also referred to as vectors. Notice that the dimension of this space is the dimension of the manifold. This linear space generates an algebra and its unit pseudoscalar characterizes the vector manifold. This is the manner in which the set of abstract vectors {x} defines the vector manifold. Once the set of “points” generates the “tangent space” the “tangent algebra” and its “pseudoscalar” follow immediately. The unit pseudoscalar of the vector manifold is a (pseudoscalar-valued) function of the points on the vector manifold. If i.e. this function is smooth then one says that the vector manifold is smooth.[3] A manifold can be defined as a set isomorphic to a vector manifold. The points of a manifold do not have any algebraic structure and pertain only to the set itself. This is the main difference between a vector manifold and a manifold that is isomorphic. A vector manifold is always a subset of Universal Geometric Algebra by definition and the elements can be manipulated algebraically. In contrast, a manifold is not a subset of any set other than itself, but the elements have no algebraic relation among them. The differential geometry of a manifold[3] can be carried out in a vector manifold. All quantities relevant to differential geometry can be calculated from In(x) if it is a differentiable function. This is the original motivation behind its definition. Vector manifolds allow an approach to the differential geometry of manifolds alternative to the “build-up” approach where structures such as metrics, connections and fiber bundles are introduced as needed.[4] The relevant structure of a vector manifold is its tangent algebra. The use of geometric calculus along with the definition of vector manifold allow the study of geometric properties of manifolds without using coordinates.

290.2 See also • Conformal geometric algebra

290.3 References [1] Pozo, José María; Sobczyk, Garret. Geometric Algebra in Linear Algebra and Geometry [2] Chapter 1 of: [D. Hestenes & G. Sobczyk] From Clifford Algebra to Geometric Calculus [3] Chapter 4 of: [D. Hestenes & G. Sobczyk] From Clifford Algebra to Geometric Calculus [4] Chapter 5 of: [D. Hestenes & G. Sobczyk] From Clifford Algebra to Geometric Calculus

• D. Hestenes, G. Sobczyk. Clifford Algebra to Geometric Calculus: a Unified Language for mathematics and Physics. Springer. ISBN 902-772-561-6. • C. Doran, A. Lasenby. “6.5 Embedded Surfaces and Vector Manifolds”. Geometric Algebra for Physicists. Cambridge University Press. ISBN 0-521-715-954. • L. Dorst, J. Lasenby (2011). “19”. Guide to Geometric Algebra in Practice. Springer. ISBN 0-857-298-100. • Hongbo Li. Invariant Algebras And Geometric Reasoning. World Scientific. ISBN 981-270-808-1. • D. Hestenes (1988). “Universal Geometric Algebra”. Quarterly Journal of Pure and Applied Mathematics 62 (3–4).

Chapter 291

Valuation ring In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x −1 belongs to D. Given a field F, if D is a subring of F such that either x or x −1 belongs to D for every nonzero x in F, then D is said to be a valuation ring for the field F or a place of F. Since F in this case is indeed the field of fractions of D, a valuation ring for a field is a valuation ring. Another way to characterize the valuation rings of a field F is that valuation rings D of F have F as their field of fractions, and their ideals are totally ordered by inclusion; or equivalently their principal ideals are totally ordered by inclusion. In particular, every valuation ring is a local ring. The valuation rings of a field are the maximal elements of the set of the local subrings in the field partially ordered by dominance or refinement,[1] where (A, mA ) dominates (B, mB ) if A ⊃ B and mA ∩ B = mB .[2] Every local ring in a field K is dominated by some valuation ring of K. An integral domain whose localization at any prime ideal is a valuation ring is called a Prüfer domain.

291.1 Examples • Any field is a valuation ring. • Z₍p₎, the localization of the integers Z at the prime ideal (p), consisting of ratios where the numerator is any integer and the denominator is not divisible by p. The field of fractions is the field of rational numbers Q. • The ring of meromorphic functions on the entire complex plane which have a Maclaurin series (Taylor series expansion at zero) is a valuation ring. The field of fractions are the functions meromorphic on the whole plane. If f does not have a Maclaurin series then 1/f does. • Any ring of p-adic integers Zp for a given prime p is a local ring, with field of fractions the p-adic numbers Q . The integral closure Zpcl of the p-adic integers is also a local ring, with field of fractions Qpcl (the algebraic closure of p-adic numbers). Both Zp and Zpcl are valuation rings. • Let k be an ordered field. An element of k is called finite if it lies between two integers n 1 using factorization over the complex numbers. In the polynomial identity

Φn (x) =

∏ (x − ζ)

where ζ runs over the primitive n-th roots of unity, set x to be q and then take absolute values

|Φn (q)| =

∏

|q − ζ|

For n > 1,

|q − ζ| > |q − 1| by looking at the location of q, 1, and ζ in the complex plane. Thus

|Φn (q)| > q − 1

294.4 Notes [1] Shult, Ernest E. (2011). Points and lines. Characterizing the classical geometries. Universitext. Berlin: Springer-Verlag. p. 123. ISBN 978-3-642-15626-7. Zbl 1213.51001. [2] Lam (2001), p. 204 [3] Theorem 4.1 in Ch. IV of Milne, class field theory, http://www.jmilne.org/math/CourseNotes/cft.html [4] e.g., Exercise 1.9 in Milne, group theory, http://www.jmilne.org/math/CourseNotes/GT.pdf

294.5. REFERENCES

877

294.5 References • Parshall, K. H. (1983). “In pursuit of the finite division algebra theorem and beyond: Joseph H M Wedderburn, Leonard Dickson, and Oswald Veblen”. Archives of International History of Science 33: 274–99. • Lam, Tsit-Yuen (2001). A first course in noncommutative rings. Graduate texts in mathematics 131 (2 ed.). Springer. ISBN 0-387-95183-0.

294.6 External links • Proof of Wedderburn’s Theorem at Planet Math

Chapter 295

Weyl algebra In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), n−1 n fn (X)∂X + fn−1 (X)∂X + · · · + f1 (X)∂X + f0 (X).

More precisely, let F be a field, and let F[X] be the ring of polynomials in one variable, X, with coefficients in F. Then each fi lies in F[X]. ∂X is the derivative with respect to X. The algebra is generated by X and ∂X. The Weyl algebra is an example of a simple ring that is not a matrix ring over a division ring. It is also a noncommutative example of a domain, and an example of an Ore extension. The Weyl algebra is a quotient of the free algebra on two generators, X and Y, by the ideal generated by elements of the form Y X − XY − 1. The Weyl algebra is the first in an infinite family of algebras, also known as Weyl algebras. The n-th Weyl algebra, An, is the ring of differential operators with polynomial coefficients in n variables. It is generated by Xi and ∂Xi . Weyl algebras are named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics. It is a quotient of the universal enveloping algebra of the Heisenberg algebra, the Lie algebra of the Heisenberg group, by setting the element 1 of the Lie algebra equal to the unit 1 of the universal enveloping algebra. The Weyl algebra is also referred to as the symplectic Clifford algebra.[1][2][3] Weyl algebras represent the same structure for symplectic bilinear forms that Clifford algebras represent for non-degenerate symmetric bilinear forms.[1]

295.1 Generators and relations One may give an abstract construction of the algebras An in terms of generators and relations. Start with an abstract vector space V (of dimension 2n) equipped with a symplectic form ω. Define the Weyl algebra W(V) to be W (V ) := T (V )/((v ⊗ u − u ⊗ v − ω(v, u), for v, u ∈ V )), where T(V) is the tensor algebra on V, and the notation (()) means “the ideal generated by”. In other words, W(V) is the algebra generated by V subject only to the relation vu − uv = ω(v, u). Then, W(V) is isomorphic to An via the choice of a Darboux basis for ω.

295.1.1

Quantization

The algebra W(V) is a quantization of the symmetric algebra Sym(V). If V is over a field of characteristic zero, then W(V) is naturally isomorphic to the underlying vector space of the symmetric algebra Sym(V) equipped with 878

295.2. PROPERTIES OF THE WEYL ALGEBRA

879

a deformed product – called the Groenewold–Moyal product (considering the symmetric algebra to be polynomial functions on V*, where the variables span the vector space V, and replacing iℏ in the Moyal product formula with 1). The isomorphism is given by the symmetrization map from Sym(V) to W(V):

a1 · · · an 7→

1 ∑ aσ(1) ⊗ · · · ⊗ aσ(n) . n! σ∈Sn

If one prefers to have the iℏ and work over the complex numbers, one could have instead defined the Weyl algebra above as generated by Xi and iℏ∂Xi (as is frequently done in quantum mechanics). Thus, the Weyl algebra is a quantization of the symmetric algebra, which is essentially the same as the Moyal quantization (if for the latter one restricts to polynomial functions), but the former is in terms of generators and relations (considered to be differential operators) and the latter is in terms of a deformed multiplication. In the case of exterior algebras, the analogous quantization to the Weyl one is the Clifford algebra, which is also referred to as the orthogonal Clifford algebra.[2][4]

295.2 Properties of the Weyl algebra In the case that the ground field F has characteristic zero, the nth Weyl algebra is a simple Noetherian domain. It has global dimension n, in contrast to the ring it deforms, Sym(V), which has global dimension 2n. It has no finite-dimensional representations; although this follows from simplicity, it can be more directly shown by taking the trace σ(X) and σ(Y) for some finite-dimensional representation σ (where [X,Y] = 1).

tr([σ(X), σ(Y )]) = tr(1) Since the trace of a commutator is zero, and the trace of the identity is the dimension of the matrix, the representation must be zero dimensional. In fact, there are stronger statements than the absence of finite-dimensional representations. To any finitely generated An-module M, there is a corresponding subvariety Char(M) of V × V* called the 'characteristic variety' whose size roughly corresponds to the size of M (a finite-dimensional module would have zero-dimensional characteristic variety). Then Bernstein’s inequality states that for M non-zero,

dim(char(M )) ≥ n An even stronger statement is Gabber’s theorem, which states that Char(M) is a co-isotropic subvariety of V × V* for the natural symplectic form.

295.2.1

Positive characteristic

The situation is considerably different in the case of a Weyl algebra over a field of characteristic p > 0. In this case, for any element D of the Weyl algebra, the element Dp is central, and so the Weyl algebra has a very large center. In fact, it is a finitely-generated module over its center; even more so, it is an Azumaya algebra over its center. As a consequence, there are many finite-dimensional representations which are all built out of simple representations of dimension p.

295.3 Generalizations For more details about this quantization in the case n = 1 (and an extension using the Fourier transform to a class of integrable functions larger than the polynomial functions), see Wigner–Weyl transform. Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras.

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295.4 References • M. Rausch de Traubenberg, M. J. Slupinski, A. Tanasa, Finite-dimensional Lie subalgebras of the Weyl algebra, (2005) (Classifies subalgebras of the one-dimensional Weyl algebra over the complex numbers; shows relationship to SL(2,C)) • Tsit Yuen Lam, A first course in noncommutative rings. Volume 131 of Graduate texts in mathematics. 2ed. Springer, 2001. p. 6. ISBN 978-0-387-95325-0 • S. C. Coutinho, The many avatars of a simple algebra. American Mathematical Monthly, Vol. 104, (1997), pp. 593-604 [1] Jacques Helmstetter, Artibano Micali: Quadratic Mappings and Clifford Algebras, Birkhäuser, 2008, ISBN 978-3-76438605-4 p. xii [2] Rafał Abłamowicz: Clifford algebras: applications to mathematics, physics, and engineering (dedicated to Pertti Lounesto), Progress in Mathematical Physics, Birkhäuser Boston, 2004, ISBN 0-8176-3525-4. Foreword, p. xvi [3] Z. Oziewicz, Cz. Sitarczyk: Parallel treatment of Riemannian and symplectic Clifford algebras, pp.83-96. In: Artibano Micali, Roger Boudet, Jacques Helmstetter (eds.): Clifford algebras and their applications in mathematical physics, Kluwer, 1989, ISBN 0-7923-1623-1, p. 92 [4] Z. Oziewicz, Cz. Sitarczyk: Parallel treatment of Riemannian and symplectic Clifford algebras, pp.83-96. In: Artibano Micali, Roger Boudet, Jacques Helmstetter (eds.): Clifford algebras and their applications in mathematical physics, Kluwer, 1989, ISBN 0-7923-1623-1, p. 83

Chapter 296

Witt vector In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors over the finite field of order p is the ring of p-adic integers.

296.1 Motivation Any p -adic integer (an element of Zp ) can be written as a power series a0 + a1 p1 + a2 p2 + · · · , where the a 's are usually taken from the set {0, 1, 2, ..., p − 1} . However, it is hard to figure out an algebraic expression for addition and multiplication, as one faces the problem of carry. Luckily, this set of representatives is not the only possible choice, and Teichmüller suggested an alternative set consisting of 0 together with the p − 1 st roots of 1 : in other words, the p roots of xp − x = 0 in Zp . These Teichmüller representatives can be identified with the elements of the finite field Fp of order p (by taking × × residues modulo p ), and elements of F× p are taken to their representatives by the Teichmüller character ω : Fp → Zp × . This identifies the set of p -adic integers with infinite sequences of elements of ω(Fp ) ∪ {0} . We now have the following problem: given two infinite sequences of elements of ω(F× p ) ∪ {0} , describe their sum and product as p -adic integers explicitly. This problem was solved by Witt using Witt vectors.

296.1.1

Details

We basically want to derive the ring p -adic integers Zp from the finite field with p elements, Fp , by some general construction. A p -adic integer is a sequence (n0 , n1 , ...) with ni ∈ Z/p(i+1) Z ,such that ni ≡ nj mod pi if i < j . They can be expanded as a power series in p : a0 + a1 p1 + a2 p2 + · · · , where the a 's are usually taken from the set {0, 1, 2, ..., p − 1} ∏ (The equation is happening in Zp , with ai and pj all images from Z to Zp ). Set-theoretically it is Fp . But Zp and N Fp are not isomorphic as rings. If we denote a + b = c , then the addition should instead be: c0 ≡ a0 + b0

mod p

c0 + c1 p ≡ a0 + a1 p + b0 + b1 p mod p2 c0 + c1 p + c2 p2 ≡ a0 + a1 p + a2 p2 + b0 + b1 p + b2 p2

mod p3

But we lack some properties of the coefficients to produce a general formula. Luckily, there is an alternative subset of Zp which can work as the coefficient set. This is the set of Teichmüller representatives of elements of Fp . Without 0 they form a subgroup of Z∗p , identified with F∗p through the Teichmüller 881

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CHAPTER 296. WITT VECTOR

character ω : F∗p → Z∗p . Note that ω is not additive, as the sum need not be a representative. Despite this, if ω(k) = ω(i) + ω(j) mod p in Zp , then i + j = k in Fp . This is conceptually justified by m ◦ ω = idFp if we denote m : Zp → Zp /pZp ∼ = Fp . Teichmüller representatives are explicitly calculated as roots of xp−1 − 1 = 0 through Hensel lifting. For example, in Z3 , to calculate the representative of 2 , you first find the unique solution of x2 − 1 = 0 in Z/9Z with x ≡ 2 mod 3 ; you get 8 , then repeat it in Z/27Z , with conditions x2 − 1 = 0 and x ≡ 2 mod 9 ; this time it is 26 , and so on. The existence of lift in each step is guaranteed by (xp−1 − 1, (p − 1)xp−2 ) = 1 in every Z/pn Z . We can also write the representatives as a0 + a1 p1 + a2 p2 + ... . Note for every j ∈ {0, 1, 2, ..., p − 1} , there is exactly one representative, namely ω(j) , with a0 = j , so we can also expand every p -adic integer as a power series in p , with coefficients from the Teichmüller representatives. Explicitly, if b = a0 + a1 p1 + a2 p2 + ... , then b − ω(a0 ) = a′1 p1 + a′2 p2 + ... . Then you subtract ω(a′1 )p and proceed similarly. Note the coefficients you get most probably differ from the ai 's modulo p , except the first one. This time we have additional properties of the coefficients like api = ai , so we can make some changes to get a neat formula. Since the Teichmüller character is not additive, we don't have c0 = a0 + b0 in Zp . But it happens in Fp , first congruence So actually cp0 ≡ ((a)0 + b0 )p mod p2 , thus c0 − a0 − b0 ≡ (a0 + b0 )p − a0 − b0 ≡ (p)implies. (asp)thep−1 p−1 b0 + ... + 1 a0 b0 mod p2 . Since pi is divisible by p , this resolves the p -coefficient problem of c1 1 a0 p−1 p−2 2 and gives c1 ≡ a1 + b1 − a0 b0 − p−1 b0 − ... − a0 bp−1 mod p . Note this completely determines c1 by the 0 2 a0 lift. Moreover, the mod p indicates that the calculation can actually be done in Fp , satisfying our basic aim. p−2 2 b0 − ... − a0 bp−1 )p Now for c2 . It is already very cumbersome at this step. c1 = cp1 ≡ (a1 + b1 − ap−1 b0 − p−1 0 0 2 a0 ( 2) 2 p2 mod p . As for c0 , a single p th power is not enough: actually we take c0 = c0 ≡ (a0 + b0 )p . pi is not always 2 divisible by p2 , but that only happens when i = pd , in which case ai bp −i = ad bp−d combined with similar p 2 monomials in c1 would make a multiple of p .

At this step, we see that we are actually working with something like

c0 ≡ a0 + b0

mod p

cp0 + c1 p ≡ ap0 + a1 p + bp0 + b1 p 2

mod p2

2

2

cp0 + cp1 p + c2 p2 ≡ ap0 + ap1 p + a2 p2 + b0p + bp1 p + b2 p2

mod p3

This motivates the definition of Witt vectors.

296.2 Construction of Witt rings Fix a prime number p. A Witt vector over a commutative ring R is a sequence : (X0 , X1 , X2 , ...) of elements of R. Define the Witt polynomials Wi by 1. W0 = X0 2. W1 = X0p + pX1 2

3. W2 = X0p + pX1p + p2 X2 and in general

Wn =

∑

pi Xip

n−i

.

i

(W0 , W1 , W2 , ...) is called the ghost components of the Witt vector (X0 , X1 , X2 , ...) , and is usually denoted by (X (0) , X (1) , X (2) , ...) . Then Witt showed that there is a unique way to make the set of Witt vectors over any commutative ring R into a ring, called the ring of Witt vectors, such that

296.3. EXAMPLES

883

• the sum and product are given by polynomials with integral coefficients that do not depend on R, and • Every Witt polynomial is a homomorphism from the ring of Witt vectors over R to R. In other words, if • (X + Y )i and (XY )i are given by polynomials with integral coefficients that do not depend on R, and • X (i) + Y (i) = (X + Y )(i) , X (i) Y (i) = (XY )(i) . The first few polynomials giving the sum and product of Witt vectors can be written down explicitly. For example, • (X0 , X1 , ...) + (Y0 , Y1 , ...) = (X0 + Y0 , X1 + Y1 + (X0p + Y0p − (X0 + Y0 )p )/p, ...) • (X0 , X1 , ...) × (Y0 , Y1 , ...) = (X0 Y0 , X0p Y1 + X1 Y0p + pX1 Y1 , ...) .

296.3 Examples • The Witt ring of any commutative ring R in which p is invertible is just isomorphic to RN (the product of a countable number of copies of R). In fact the Witt polynomials always give a homomorphism from the ring of Witt vectors to RN , and if p is invertible this homomorphism is an isomorphism. • The Witt ring of the finite field of order p is the ring of p-adic integers, as is demonstrated above. • The Witt ring of a finite field of order pn is the unramified extension of degree n of the ring of p-adic integers.

296.4 Universal Witt vectors The Witt polynomials for different primes p are special cases of universal Witt polynomials, which can be used to form a universal Witt ring (not depending on a choice of prime p). Define the universal Witt polynomials Wn for n≥1 by 1. W1 = X1 2. W2 = X12 + 2X2 3. W3 = X13 + 3X3 4. W4 = X14 + 2X22 + 4X4 and in general

Wn =

∑

n/d

dXd

.

d|n

Again, (W1 , W2 , W3 , ...) is called the ghost components of the Witt vector (X1 , X2 , X3 , ...) , and is usually denoted by (X (1) , X (2) , X (3) , ...) . We can use these polynomials to define the ring of universal Witt vectors over any commutative ring R in much the same way as above (so the universal Witt polynomials are all homomorphisms to the ring R).

296.5 Generating Functions Later Witt orally stated another approach using generating functions.[1]

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CHAPTER 296. WITT VECTOR

296.5.1

Definition

Let X be a Witt vector and define

fX (t) =

∏

∑

(1 − Xn tn ) =

n≥1

A n tn

n≥0

For let∑ Sn denote the collection of subsets of {1, 2, ..., n} whose elements add up to n . Then An = ∑ n ≥ 1 |S| S∈S (−1) i∈S Xi . We can get the ghost components by taking the logarithmic derivative:

∑ d ∑ ∑ X d tnd ∑ d n log fX (t) = (1 − Xn tn ) = − =− dt dt d n≥1

296.5.2

n≥1 d≥1

m≥1

∑

m d d|m d X m m d

m

t

=−

∑ X (m) tm m

m≥1

Sum

∑ Now we can see fZ (t) = fX (t)fY (t) if Z = X +Y . So that Cn = 0≤i≤n An Bn−i if An , Bn , Cn are respective ∑ ∑ ∑ coefficients in the power series for fX (t), fY (t), fZ (t) . Then Zn = 0≤i≤n An Bn−i − S∈S,S̸={n} (−1)|S| i∈S Zi . Since An is a polynomial in X1 , ..., Xn and likely for Bn , we can show by induction that Zn is a polynomial in X1 , ..., Xn , Y1 , ..., Yn .

296.5.3

Product

If we set W = XY then ∑ X (m) Y (m) tm d log fW (t) = − dt m m≥1

But

∑ X (m) Y (m) ∑ tm = m

m≥1

∑

m≥1

m/d

d|m

dXd

∑ e|m

m/e

eYe

m

tm

Now 3-tuples m, d, e with m ∈ Z+ , d|m, e|m are in bijection with 3-tuples d, e, n with d, e, n ∈ Z+ , via n = m/[d, e] ( [d, e] is the Least common multiple), our series becomes

∑

de [d,e]

∑

[d,e]/d [d,e]/e [d,e] n Ye t ) n≥1 (Xd

n

d,e≥1

So that

fW (t) =

∏

[d,e]/d

(1 − Xd

d,e≥1

Ye[d,e]/e t[d,e] )de/[d,e] =

∑

Dn tn

n≥0

where Dn s are polynomials of X1 , ..., Xn , Y1 , ..., Yn . So by similar induction, suppose fW (t) = , then Wn can be solved as polynomials of X1 , ..., Xn , Y1 , ..., Yn .

∏

n≥1 (1−Wn t

n

)

296.6. RING SCHEMES

885

296.6 Ring schemes The map taking a commutative ring R to the ring of Witt vectors over R (for a fixed prime p) is a functor from commutative rings to commutative rings, and is also representable, so it can be thought of as a ring scheme, called the Witt scheme, over Spec(Z). The Witt scheme can be canonically identified with the spectrum of the ring of symmetric functions. Similarly the rings of truncated Witt vectors, and the rings of universal Witt vectors, correspond to ring schemes, called the truncated Witt schemes and the universal Witt scheme. Moreover, the functor taking the commutative ring R to the set Rn is represented by the affine space AnZ , and the ring structure on Rn makes AnZ into a ring scheme denoted On . From the construction of truncated Witt vectors it follows that their associated ring scheme Wn is the scheme AnZ with the unique ring structure such that the morphism Wn → On given by the Witt polynomials is a morphism of ring schemes.

296.7 Commutative unipotent algebraic groups Over an algebraically closed field of characteristic 0, any unipotent abelian connected algebraic group is isomorphic to a product of copies of the additive group Ga . The analogue of this for fields of characteristic p is false: the truncated Witt schemes are counterexamples. (We make them into algebraic groups by forgetting the multiplication and just using the additive structure.) However these are essentially the only counterexamples: over an algebraically closed field of characteristic p, any unipotent abelian connected algebraic group is isogenous to a product of truncated Witt group schemes.

296.8 See also • Formal group • Artin–Hasse exponential

296.9 References [1] Lang, Serge (September 19, 2005). “Chapter VI: Galois Theory”. Algebra (3rd ed.). Springer. p. 330. ISBN 978-0-38795385-4.

• Dolgachev, I.V. (2001), “Witt vector”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Hazewinkel, Michiel (2009), “Witt vectors. I.”, Handbook of algebra. Vol. 6, Amsterdam: Elsevier/NorthHolland, pp. 319–472, arXiv:0804.3888, ISBN 978-0-444-53257-2, MR 2553661 • Mumford, David, Lectures on Curves on an Algebraic Surface, Annals of Mathematics Studies 59, Princeton, NJ: Princeton University Press, ISBN 978-0-691-07993-6 • Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics 67, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90424-5, MR 554237, section II.6 • Serre, Jean-Pierre (1988), Algebraic groups and class fields, Graduate Texts in Mathematics 117, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96648-9, MR 918564 • Witt, Ernst (1936), “Zyklische Körper und Algebren der Characteristik p vom Grad pn . Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik pn ", Journal für Reine und Angewandte Mathematik (in German) 176: 126–140, doi:10.1515/crll.1937.176.126 • Greenberg, M. J. (1969), Lectures on Forms in Many Variables, New York and Amsterdam, Benjamin, MR 241358, ASIN: B0006BX17M

Chapter 297

Zero divisor In abstract algebra, an element a of a ring R is called a left zero divisor if there exists a nonzero x such that ax = 0,[1] or equivalently if the map from R to R that sends x to ax is not injective.[2] Similarly, an element a of a ring is called a right zero divisor if there exists a nonzero y such that ya = 0. This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor.[3] An element a that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero x such that ax = 0 may be different from the nonzero y such that ya = 0). If the ring is commutative, then the left and right zero divisors are the same. An element of a ring that is not a zero divisor is called regular, or a non-zero-divisor. A zero divisor that is nonzero is called a nonzero zero divisor or a nontrivial zero divisor.

297.1 Examples • In the ring Z/4Z , the residue class 2 is a zero divisor since 2 × 2 = 4 = 0 . • The only zero divisor of the ring Z of integers is 0. • A nilpotent element of a nonzero ring is always a two-sided zero divisor. • A idempotent element e ̸= 1 of a ring is always a two-sided zero divisor, since e(1 − e) = 0 = (1 − e)e . • Examples of zero divisors in the ring of 2 × 2 matrices (over any nonzero ring) are shown here: ( )( ) ( )( ) ( ) 1 1 1 1 −2 1 1 1 0 0 = = , 2 2 −1 −1 −2 1 2 2 0 0 ( 1 0

0 0

)( ) ( )( 0 0 0 0 1 = 0 1 0 1 0

0 0

)

( =

0 0 0 0

)

• A direct product of two or more nonzero rings always has nonzero zero divisors. For example, in R1 × R2 with each Ri nonzero, (1,0)(0,1) = (0,0), so (1,0) is a zero divisor.

297.1.1

One-sided zero-divisor

( ) ( )( ) x y x y a b • Consider the ring of (formal) matrices with x, z ∈ Z and y ∈ Z/2Z . Then = z 0 c ( ) ( )( ) 0 (z ) ( 0 ) xa xb + yc a b x y xa ya + zb x y and = . If x ̸= 0 ̸= y , then is a left zero 0 zc 0 (c 0)(z 0 zc 0 z ) ( ) x y 0 1 0 x = ; and it is a right zero divisor iff z is even for similar divisor iff x is even, since 0 z 0 0 0 0 reasons. If either of x, z is 0 , then it is a two-sided zero-divisor. • Here is another example of a ring with an element that is a zero divisor on one side only. Let S be the set of all sequences of integers (a1, a2, a3, ...) . Take for the ring all additive maps from S to S , with pointwise 886

297.2. NON-EXAMPLES

887

addition and composition as the ring operations. (That is, our ring is End(S) , the endomorphism ring of the additive group S .) Three examples of elements of this ring are the right shift R(a1, a2, a3, ...) = (0, a1, a2, ...) , the left shift L(a1, a2, a3, ...) = (a2, a3, a4, ...) , and the projection map onto the first factor P (a1, a2, a3, ...) = (a1, 0, 0, ...) . All three of these additive maps are not zero, and the composites LP and P R are both zero, so L is a left zero divisor and R is a right zero divisor in the ring of additive maps from S to S . However, L is not a right zero divisor and R is not a left zero divisor: the composite LR is the identity. Note also that RL is a two-sided zero-divisor since RLP = 0 = P RL , while LR = 1 is not in any direction.

297.2 Non-examples • The ring of integers modulo a prime number has no zero divisors other than 0. Since every nonzero element is a unit, this ring is a field. • More generally, a division ring has no zero divisors except 0. • A nonzero commutative ring whose only zero divisor is 0 is called an integral domain.

297.3 Properties • In the ring of n-by-n matrices over a field, the left and right zero divisors coincide; they are precisely the singular matrices. In the ring of n-by-n matrices over an integral domain, the zero divisors are precisely the matrices with determinant zero. • Left or right zero divisors can never be units, because if a is invertible and ax = 0, then 0 = a−1 0 = a−1 ax = x, whereas x must be nonzero.

297.4 Zero as a zero divisor There is no need for a separate convention regarding the case a = 0, because the definition applies also in this case: • If R is a ring other than the zero ring, then 0 is a (two-sided) zero divisor, because 0 · 1 = 0 and 1 · 0 = 0. • If R is the zero ring, in which 0 = 1, then 0 is not a zero divisor, because there is no nonzero element that when multiplied by 0 yields 0. Such properties are needed in order to make the following general statements true: • In a commutative ring R, the set of non-zero-divisors is a multiplicative set in R. (This, in turn, is important for the definition of the total quotient ring.) The same is true of the set of non-left-zero-divisors and the set of non-right-zero-divisors in an arbitrary ring, commutative or not. • In a commutative Noetherian ring R, the set of zero divisors is the union of the associated prime ideals of R. Some references choose to exclude 0 as a zero divisor by convention, but then they must introduce exceptions in the two general statements just made.

297.5 Zero divisor on a module Let R be a commutative ring, let M be an R-module, and let a be an element of R. One says that a is M-regular if a the multiplication by a map M →M is injective, and that a is a zero divisor on M otherwise.[4] The set of M-regular elements is a multiplicative set in R.[5] Specializing the definitions of “M-regular” and “zero divisor on M” to the case M = R recovers the definitions of “regular” and “zero divisor” given earlier in this article.

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CHAPTER 297. ZERO DIVISOR

297.6 See also • Zero-product property • Glossary of commutative algebra (Exact zero divisor)

297.7 Notes [1] See Bourbaki, p. 98. [2] Since the map is not injective, we have ax = ay, in which x differs from y, and thus a(x-y) = 0. [3] See Lanski (2005). [4] Matsumura, p. 12 [5] Matsumura, p. 12

297.8 References • N. Bourbaki (1989), Algebra I, Chapters 1–3, Springer-Verlag. • Hazewinkel, Michiel, ed. (2001), “Zero divisor”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608010-4 • Michiel Hazewinkel; Nadiya Gubareni; Nadezhda Mikhaĭlovna Gubareni; Vladimir V. Kirichenko. (2004), Algebras, rings and modules, Vol. 1, Springer, ISBN 1-4020-2690-0 • Charles Lanski (2005), Concepts in Abstract Algebra, American Mathematical Soc., p. 342 • Hideyuki Matsumura (1980), Commutative algebra, 2nd edition, The Benjamin/Cummings Publishing Company, Inc. • Weisstein, Eric W., “Zero Divisor”, MathWorld.

Chapter 298

Zero object (algebra) This article is about zero objects or trivial objects in algebraic structures. For zero object in a category, see Initial and terminal objects. For trivial representation, see Trivial representation. In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such

0 {0} A

A′ Morphisms to and from the zero object

structure. As a set it is a singleton, and also has a trivial structure of abelian group. Aforementioned group structure usually identified as the addition, and the only element is called zero 0, so the object itself is denoted as {0}. One often refers to the trivial object (of a specified category) since every trivial object is isomorphic to any other (under a unique isomorphism). Instances of the zero object include, but are not limited to the following: • As a group, the trivial group. • As a ring, the trivial ring. • As a module (over a ring R), the zero module. The term trivial module is also used, although it is ambiguous. • As a vector space (over a field R), the zero vector space, zero-dimensional vector space or just zero space. • As an algebra over a field or algebra over a ring, the trivial algebra. These objects are described jointly not only based on the common singleton and trivial group structure, but also because of shared category-theoretical properties. 889

890

CHAPTER 298. ZERO OBJECT (ALGEBRA)

In the last three cases the scalar multiplication by an element of the base ring (or field) is defined as: κ0 = 0 , where κ ∈ R. The most general of them, the zero module, is a finitely-generated module with an empty generating set. For structures requiring the multiplication structure inside the zero object, such as the trivial ring, there is only one possible, 0 × 0 = 0, because there are no non-zero elements. This structure is associative and commutative. A ring R which has both an additive and multiplicative identity is trivial if and only if 1 = 0, since this equality implies that for all r within R,

r = r × 1 = r × 0 = 0. In this case it is possible to define division by zero, since the single element is its own multiplicative inverse. Some properties of {0} depend on exact definition of the multiplicative identity; see the section Unital structures below. Any trivial algebra is also a trivial ring. A trivial algebra over a field is simultaneously a zero vector space considered below. Over a commutative ring, a trivial algebra is simultaneously a zero module. The trivial ring is an example of a pseudo-ring of square zero. A trivial algebra is an example of a zero algebra. The zero-dimensional vector space is an especially ubiquitous example of a zero object, a vector space over a field with an empty basis. It therefore has dimension zero. It is also a trivial group over addition, and a trivial module mentioned above.

298.1 Properties The trivial ring, zero module and zero vector space are zero objects of the corresponding categories, namely Rng, R-Mod and VectR. The zero object, by definition, must be a terminal object, which means that a morphism A → {0} must exist and be unique for an arbitrary object A. This morphism maps any element of A to 0. The zero object, also by definition, must be an initial object, which means that a morphism {0} → A must exist and be unique for an arbitrary object A. This morphism maps 0, the only element of {0}, to the zero element 0 ∈ A, called the zero vector in vector spaces. This map is a monomorphism, and hence its image is isomorphic to {0}. For modules and vector spaces, this subset {0} ⊂ A is the only empty-generated submodule (or 0-dimensional linear subspace) in each module (or vector space) A.

298.1.1

Unital structures

The {0} object is a terminal object of any algebraic structure where it exists, like it was described for examples above. But its existence and, if it exists, the property to be an initial object (and hence, a zero object in the category-theoretical sense) depend on exact definition of the multiplicative identity 1 in a specified structure. If the definition of 1 requires that 1 ≠ 0, then the {0} object cannot exist because it may contain only one element. In particular, the zero ring is not a field. If mathematicians sometimes talk about a field with one element, this abstract and somewhat mysterious mathematical object is not a field. In categories where the multiplicative identity must be preserved by morphisms, but can equal to zero, the {0} object can exist. But not as initial object because identity-preserving morphisms from {0} to any object where 1 ≠ 0 do not exist. For example, in the category of rings Ring the ring of integers Z is the initial object, not {0}. If an algebraic structure requires the multiplicative identity, but does not require neither its preserving by morphisms nor 1 ≠ 0, then zero morphisms exist and the situation is not different from non-unital structures considered in the previous section.

298.2. NOTATION

891

298.2 Notation Zero vector spaces and zero modules are usually denoted by 0 (instead of {0}). This is always the case when they occur in an exact sequence.

298.3 See also • Triviality (mathematics) • Examples of vector spaces • Field with one element • Empty semigroup • Zero element (disambiguation) • List of zero terms

298.4 External links • David Sharpe (1987). Rings and factorization. Cambridge University Press. p. 10 : trivial ring. ISBN 0-52133718-6. • Barile, Margherita, “Trivial Module”, MathWorld. • Barile, Margherita, “Zero Module”, MathWorld.

Chapter 299

Zero ring In ring theory, a branch of mathematics, the zero ring[1][2][3][4][5] or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term “zero ring” is used to refer to any rng of square zero, i.e., a rng in which xy = 0 for all x and y. This article refers to the one-element ring.) In the category of rings, the zero ring is the terminal object, whereas the ring of integers Z is the initial object.

299.1 Definition The zero ring, denoted {0} or simply 0, consists of the one-element set {0} with the operations + and · defined so that 0 + 0 = 0 and 0 · 0 = 0.

299.2 Properties • The zero ring is the unique ring in which the additive identity 0 and multiplicative identity 1 coincide.[6][7] (Proof: If 1 = 0 in a ring R, then for all r in R, we have r = 1r = 0r = 0.) • The zero ring is commutative. • The element 0 in the zero ring is a unit, serving as its own multiplicative inverse. • The unit group of the zero ring is the trivial group {0}. • The element 0 in the zero ring is not a zero divisor. • The only ideal in the zero ring is the zero ideal {0}, which is also the unit ideal, equal to the whole ring. This ideal is neither maximal nor prime. • The zero ring is not a field; this agrees with the fact that its zero ideal is not maximal. In fact, there is no field with less than 2 elements. (When mathematicians speak of the "field with one element", they are referring to a non-existent object, and their intention is to define the category that would be the category of schemes over this object if it existed.) • The zero ring is not an integral domain.[8] Whether the zero ring is considered to be a domain at all is a matter of convention, but there are two advantages to considering it not to be a domain. First, this agrees with the definition that a domain is a ring in which 0 is the only zero divisor (in particular, 0 is required to be a zero divisor, which fails in the zero ring). Second, this way, for a positive integer n, the ring Z/nZ is a domain if and only if n is prime. • For each ring A, there is a unique ring homomorphism from A to the zero ring. Thus the zero ring is a terminal object in the category of rings.[9] • If A is a nonzero ring, then there is no ring homomorphism from the zero ring to A. In particular, the zero ring is not a subring of any nonzero ring.[10] 892

299.3. CONSTRUCTIONS

893

• The characteristic of the zero ring is 1. • The only module for the zero ring is the zero module. It is free of rank ‫ א‬for any cardinal number ‫א‬. • The zero ring is not a local ring. It is, however, a semilocal ring. • The zero ring is Artinian and (therefore) Noetherian. • The spectrum of the zero ring is the empty scheme.[11] • The Krull dimension of the zero ring is −∞. • The zero ring is semisimple but not simple. • The zero ring is not a central simple algebra over any field. • The total quotient ring of the zero ring is itself.

299.3 Constructions • For any ring A and ideal I of A, the quotient A/I is the zero ring if and only if I=A, i.e. iff I is the unit ideal. • For any commutative ring A and multiplicative set S in A, the localization S −1 A is the zero ring if and only if S contains 0. • If A is any ring, then the ring M0 (A) of 0 × 0 matrices over A is the zero ring. • The direct product of an empty collection of rings is the zero ring. • The endomorphism ring of the trivial group is the zero ring. • The ring of continuous real-valued functions on the empty topological space is the zero ring.

299.4 Notes [1] Artin, p. 347. [2] Atiyah and Macdonald, p. 1. [3] Bosch, p. 10. [4] Bourbaki, p. 101. [5] Lam, p. 1. [6] Artin, p. 347. [7] Lang, p. 83. [8] Lam, p. 3. [9] Hartshorne, p. 80. [10] Hartshorne, p. 80. [11] Hartshorne, p. 80.

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CHAPTER 299. ZERO RING

299.5 References • Michael Artin, Algebra, Prentice-Hall, 1991. • Siegfried Bosch, Algebraic geometry and commutative algebra, Springer, 2012. • M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley, 1969. • N. Bourbaki, Algebra I, Chapters 1-3. • Robin Hartshorne, Algebraic geometry, Springer, 1977. • T. Y. Lam, Exercises in classical ring theory, Springer, 2003. • Serge Lang, Algebra 3rd ed., Springer, 2002.

Chapter 300

Zero-product property In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, it is the following assertion: If ab = 0 , then a = 0 or b = 0 . The zero-product property is also known as the rule of zero product, the null factor law or the nonexistence of nontrivial zero divisors. All of the number systems studied in elementary mathematics — the integers Z , the rational numbers Q , the real numbers R , and the complex numbers C — satisfy the zero-product property. In general, a ring which satisfies the zero-product property is called a domain.

300.1 Algebraic context Suppose A is an algebraic structure. We might ask, does A have the zero-product property? In order for this question to have meaning, A must have both additive structure and multiplicative structure.[note 1] Usually one assumes that A is a ring, though it could be something else, e.g., the nonnegative integers {0, 1, 2, . . .} . Note that if A satisfies the zero-product property, and if B is a subset of A , then B also satisfies the zero product property: if a and b are elements of B such that ab = 0 , then either a = 0 or b = 0 because a and b can also be considered as elements of A .

300.2 Examples • A ring in which the zero-product property holds is called a domain. A commutative domain with a multiplicative identity element is called an integral domain. Any field is an integral domain; in fact, any subring of a field is an integral domain (as long as it contains 1). Similarly, any subring of a skew field is a domain. Thus, the zero-product property holds for any subring of a skew field. • If p is a prime number, then the ring of integers modulo p has the zero-product property (in fact, it is a field). • The Gaussian integers are an integral domain because they are a subring of the complex numbers. • In the strictly skew field of quaternions, the zero-product property holds. This ring is not an integral domain, because the multiplication is not commutative. • The set of nonnegative integers {0, 1, 2, . . .} is not a ring, but it does satisfy the zero-product property. 895

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CHAPTER 300. ZERO-PRODUCT PROPERTY

300.3 Non-examples • Let Zn denote the ring of integers modulo n . Then Z6 does not satisfy the zero product property: 2 and 3 are nonzero elements, yet 2 · 3 ≡ 0 (mod 6) . • In general, if n is a composite number, then Zn does not satisfy the zero-product property. Namely, if n = qm where 0 < q, m < n , then m and q are nonzero modulo n , yet qm ≡ 0 (mod n) . • The ring Z2×2 of 2 by 2 matrices with integer entries does not satisfy the zero-product property: if ( 1 M= 0

) ( ) −1 0 1 and N = , 0 0 1

then

( MN =

)( ) ( 1 −1 0 1 0 = 0 0 0 1 0

0 0

) =0

yet neither M nor N is zero. • The ring of all functions f : [0, 1] → R , from the unit interval to the real numbers, has nontrivial zero divisors: there are pairs of functions which are not identically equal to zero yet whose product is the zero function. In fact, it is not hard to construct, for any n ≥ 2, functions f1 , . . . , fn , none of which is identically zero, such that fi fj is identically zero whenever i ̸= j . • The same is true even if we consider only continuous functions, or only even infinitely smooth functions.

300.4 Application to finding roots of polynomials Suppose P and Q are univariate polynomials with real coefficients, and x is a real number such that P (x)Q(x) = 0 . (Actually, we may allow the coefficients and x to come from any integral domain.) By the zero-product property, it follows that either P (x) = 0 or Q(x) = 0 . In other words, the roots of P Q are precisely the roots of P together with the roots of Q . Thus, one can use factorization to find the roots of a polynomial. For example, the polynomial x3 − 2x2 − 5x + 6 factorizes as (x − 3)(x − 1)(x + 2) ; hence, its roots are precisely 3, 1, and −2. In general, suppose R is an integral domain and f is a monic univariate polynomial of degree d ≥ 1 with coefficients in R . Suppose also that f has d distinct roots r1 , . . . , rd ∈ R . It follows (but we do not prove here) that f factorizes as f (x) = (x − r1 ) · · · (x − rd ) . By the zero-product property, it follows that r1 , . . . , rd are the only roots of f : any root of f must be a root of (x − ri ) for some i . In particular, f has at most d distinct roots. If however R is not an integral domain, then the conclusion need not hold. For example, the cubic polynomial x3 + 3x2 + 2x has six roots in Z6 (though it has only three roots in Z ).

300.5 See also • Fundamental theorem of algebra • Integral domain and domain • Prime ideal • Zero divisor

300.6. NOTES

300.6 Notes [1] There must be a notion of zero (the additive identity) and a notion of products, i.e., multiplication.

300.7 References • David S. Dummit and Richard M. Foote, Abstract Algebra (3d ed.), Wiley, 2003, ISBN 0-471-43334-9.

300.8 External links • PlanetMath: Zero rule of product

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Chapter 301

Zorn ring In mathematics, a Zorn ring is an alternative ring in which for every non-nilpotent x there exists an element y such that xy is a non-zero idempotent (Kaplansky 1968, pages 19, 25). Kaplansky (1951) named them after Max August Zorn, who studied a similar condition in (Zorn 1941). For associative rings, the definition of Zorn ring can be restated as follows: the Jacobson radical J(R) is a nil ideal and every right ideal of R which is not contained in J(R) contains a nonzero idempotent. Replacing “right ideal” with “left ideal” yields an equivalent definition. Left or right Artinian rings, left or right perfect rings, semiprimary rings and von Neumann regular rings are all examples of associative Zorn rings.

301.1 References • Kaplansky, Irving (1951), “Semi-simple alternative rings”, Portugaliae mathematica 10 (1): 37–50, MR 0041835 • Kaplansky, I. (1968), Rings of Operators, New York: W. A. Benjamin, Inc. • Tuganbaev, A. A. (2002), “Semiregular, weakly regular, and $\pi$-regular rings”, J. Math. Sci. (New York) 109: 1509–1588, MR 1871186 • Zorn, Max (1941), “Alternative rings and related questions I: existence of the radical”, Annals of Mathematics, Second Series 42: 676–686, JSTOR 1969256, MR 0005098

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Chapter 302

λ-ring In algebra, a λ-ring or lambda ring is a commutative ring together with some operations λn on it behaving like the exterior powers of vector spaces. Many rings considered in K-theory carry a natural λ-ring structure. λ-rings also provide a powerful formalism for studying an action of the symmetric functions on the ring of polynomials, recovering and extending many classical results (Lascoux (2003)). λ-rings were introduced by Grothendieck (1957, 1958, p.148). For more about λ-rings see Atiyah & Tall (1969), Knutson (1973), Hazewinkel (2009) and Yau (2010).

302.1 Intuition If V and W are finite-dimensional vector spaces over a field k, then we can form the direct sum V⊕W, the tensor product V⊗W, and the n-th exterior power of V, Λn (V). All of these are again finite-dimensional vector spaces over k. The same three operations of direct sum, tensor product and exterior power are also available when working with k-linear representations of a finite group, and when working with vector bundles over some topological space, and in more general situations. λ-rings are designed to abstract the common algebraic properties of these three operations, where we also allow for formal inverses with respect to the direct sum operation. The addition in the ring corresponds to the direct sum, the multiplication in the ring corresponds to the tensor product, and the λ-operations to the exterior powers. For example, the isomorphism ( ) Λ2 (V ⊕ W ) ∼ = Λ2 (V ) ⊕ Λ1 (V ) ⊗ Λ1 (W ) ⊕ Λ2 (W ) corresponds to the formula

λ2 (x + y) = λ2 (x) + λ1 (x)λ1 (y) + λ2 (y) valid in all λ-rings, and the isomorphism

Λ1 (V ⊗ W ) ∼ = Λ1 (V ) ⊗ Λ1 (W ) corresponds to the formula

λ1 (xy) = λ1 (x)λ1 (y) valid in all λ-rings. Analogous but (much) more complicated formulas govern the higher order λ-operators. 899

900

CHAPTER 302. Λ-RING

302.2 Definition A λ-ring is a commutative ring R together with operations λn :R→R for every non-negative integer n. These operations are assumed to behave like exterior powers of vector spaces, in the sense that they have the same behavior on sums and products that exterior powers have on direct sums and tensor products of vector spaces, and behave in the same way as exterior powers under composition. In more detail, they have the following properties valid for all x, y ∈ R and all n≥0: • λ0 (x) = 1 • λ1 (x) = x • λn (1) = 0 if n ≥ 2 • λn (x + y) = Σi₊j₌n λi (x)λj (y) • λn (xy) = Pn(λ1 (x), ..., λn (x), λ1 (y), ..., λn (y)) • λm (λn (x)) = Pm,n(λ1 (x), ..., λmn (x)) where Pn and Pm,n are universal polynomials with integer coefficients describing the behavior of exterior powers on tensor products and under composition, that can be described as follows. Suppose a commutative ring has elements x = x1 + x2 + ...,y = y1 + y2 + ... and define λn (x) by •

∑ m

λm (x)tm =

∏ (1 + txi ) i

and similarly for y. Informally we think of x and y as vector bundles that are sums of line bundles xi, yj, and think of λn (x) as the nth exterior power of x. Then the polynomials Pn and Pm,n are the universal polynomials such that •

∑

Pm (λ1 (x), · · · , λm (x), λ1 (y), · · · , λm (y))tm =

m

•

∑

∏ (1 + txi yj ) i,j

Pm,n (λ1 (x), · · · , λmn (x))tm =

m

∏

(1 + txi1 xi2 · · · xin ) for every integer n≥1.

i1