1 *-algebra 1 1.1 Terminology . . . 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
2 Acceptable ring 5 2.1 References. . . 5 3 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
4 Additive map 9 4.1 Additive map of a division ring . . . 9
4.2 References . . . 9
5 Algebra (ring theory) 10 5.1 Formal definition . . . 10
5.2 Example . . . 10 i
5.2.1 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 6 Algebra homomorphism 13 6.1 Unital algebra homomorphisms . . . 13
6.2 Examples . . . 13
6.3 See also . . . 14
6.4 References. . . 14
7 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
7.2 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 8 Algebra representation 20 8.1 Examples . . . 20
8.1.1 Linear complex structure . . . 20
8.1.2 Polynomial algebras . . . 20
8.2 Weights . . . 20
8.3 See also . . . 21
8.4 Notes . . . 21
8.5 References. . . 21
9 Algebraically compact module 22 9.1 Definitions . . . 22
9.2 Examples . . . 22
9.4 See also . . . 23
9.5 References . . . 23
10 Almost commutative ring 24 10.1 See also . . . 24
10.2 References . . . 24
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 . . . 33
16.3 Commutative Artinian rings . . . 33
16.4 Simple Artinian ring . . . 34
16.5 See also . . . 34
16.6 Notes . . . 34
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 . . . 40
20 Ascending chain condition on principal ideals 41 20.1 Commutative rings . . . 41
20.2 Noncommutative rings. . . 42
20.3 References . . . 42
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 48 23.1 References . . . 49 24 Azumaya algebra 50 24.1 References . . . 50 25 Baer ring 51 25.1 Definitions . . . 51 25.2 Examples . . . 52 25.3 Properties . . . 52 25.4 See also . . . 52
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 29 Binomial ring 59 29.1 References . . . 59 30 Biquaternion 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 Λ2ℝn . . . . 68
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 ℝ4 . . . . 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
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 . . . 95 36 Bézout domain 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 99 38 Cartan–Brauer–Hua theorem 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
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) 113 44.1 Other equivalent characterizations . . . 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
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
48.2.1 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 146 50.1 Basic concepts and operations . . . 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
51.2.2 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 . . . 171 54.6 Bibliography . . . 171 54.6.1 Books . . . 171 54.6.2 Online resources . . . 172 55 Connected ring 173 55.1 Examples and non-examples. . . 173
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 . . . 180
59.5 References . . . 180
60 Depth (ring theory) 182 60.1 Definition . . . 182
60.1.1 Theorem (Rees) . . . 182
60.2 Depth and projective dimension . . . 182
60.3 Depth zero rings . . . 183
60.4 References . . . 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
62.1 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
66.5 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) 203 67.1 Examples and non-examples . . . 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
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
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
81.3.4 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
85.2.1 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
91.3 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
94.3 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 . . . 326 99 Graded ring 327 99.1 First properties . . . 327 99.2 Graded module . . . 328
99.3 Invariants of graded modules . . . 328
99.5 G-graded rings and algebras . . . 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
102.4References . . . 343 103Hermite ring 344 103.1References . . . 344 104Hilbert–Kunz function 345 104.1References . . . 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
108.9References . . . 355
109Ideal (ring theory) 356
109.1History . . . 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 . . . 363 110.6Connections to class field theory . . . 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
113.2In 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 . . . 379 117.3Theory . . . 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
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 . . . 383 118Integer 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
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 . . . 408 126Irreducible ideal 410 126.1See also . . . 410 126.2References . . . 410 127Irreducible ring 411 127.1Definitions . . . 411
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
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
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
148.3.2 General Case . . . 452 148.4Notes . . . 453 148.5References . . . 453 148.6See also . . . 453 149Localization (algebra) 454 149.1Construction . . . 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 463 151.1Terminology . . . 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
151.7References . . . 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
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 . . . 491 162Monoid ring 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
163Morita equivalence 494 163.1Motivation . . . 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 P2 . . . 500 164.3.2 Multivectors on P3 . . . 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
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
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
179.2See 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
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 . . . 561 187.2.2 Properties . . . 561 187.3Formally verified results for commutative ordered rings . . . 561 187.4References . . . 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
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 . . . 567 189.3References . . . 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
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
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 598
195.1References . . . 598
196Primary ideal 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
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 . . . 622 206.3References . . . 622 207Product of rings 624 207.1Examples . . . 624 207.2Properties . . . 624
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 633
210.1Definitions . . . 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 . . . 638 212Pseudoscalar 639
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
218Quasisymmetric function 658 218.1Definitions . . . 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
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 681
227.1Properties and examples . . . 681 227.2Relation with semihereditary rings . . . 682 227.3See also . . . 682 227.4References . . . 682
228Regev’s theorem 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
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
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 . . . 711 235.6.5 Localization . . . 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 . . . 717 235.11Generalization . . . 717 235.11.1Rng . . . 717 235.11.2Nonassociative ring . . . 718 235.11.3Semiring . . . 718 235.12Other ring-like objects . . . 718
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 729
238.1Symmetric multilinear maps . . . 729 238.2See also . . . 730 238.3Notes . . . 730 238.4References . . . 730 239Ring theory 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
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
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
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
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 . . . 780 257.4The Jacobson density theorem . . . 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
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
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
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 . . . 834
277Tight closure 835
