Download Free Algebras, Rings and Modules Non-commutative Algebras and Rings Author Michiel Hazewinkel and Nadiya Gubareni
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CONTENTS
Preface v
1. Preliminaries 1
1.1 Basic Concepts Concerning Rings and Modules 1
1.2 Categories and Functors 10
1.3 Tensor Product of Modules 15
1.4 Direct and Inverse Limits 17
1.5 Projective, Injective, and Flat Modules 19
1.6 The Functor Tor 23
1.7 The Functor Ext 24
1.8 Hereditary and Semihereditary Rings 26
1.9 Semiperfect and Perfect Rings 28
1.10 Serial and Semidistributive Rings 31
1.11 Classical Rings of Fractions 34
1.12 Quivers of Algebras and Rings 35
2. Basic General Constructions of Groups and Rings 37
2.1 Direct and Semidirect Products 37
2.2 Group Rings, Skew Group Rings and Twisted Group Rings 44
2.3 Polynomial and Skew Polynomial Rings 49
2.4 Formal Power and Skew Formal Power Series Rings 55
2.5 Laurent Polynomial Rings and Laurent Power Series Rings 62
2.6 Generalized Matrix Rings. Generalized Triangular Matrix Rings 65
2.7 G-graded Rings 70
2.8 Notes and References 73
3. Valuation Rings 74
3.1 Valuation Domains 75
3.2 Discrete Valuation Domains 85
3.3 Invariant Valuation Rings of Division Rings 87
3.4 Examples of Non-commutative Non-discretely-valued Valuation Rings 93
3.5 Discrete Valuation Rings of Division Rings 99
3.6 Total Valuation Rings 102
3.7 Other Kinds of Valuation Rings with Zero Divisors 105
3.8 Approximation Theorems for Valuation Rings 113
3.9 Notes and References 117
4. Homological Dimensions of Rings and Modules 119
4.1 Projective and Injective Dimension 120
4.2 Flat and Weak Dimension 122
4.3 Some Examples 124
4.4 Homological Characterization of Some Classes of Rings 130
4.5 Torsion-less Modules 136
4.6 Flat Modules and Coherent Rings 144
4.7 Modules over Formal Triangular Matrix Rings 152
4.8 Notes and References 158
5. Goldie and Krull Dimensions of Rings and Modules 160
5.1 Uniform Modules and Uniform Dimension 161
5.2 Injective Uniform Modules 171
5.3 Nonsingular Modules and Rings 175
5.4 Nonsingular Rings and Goldie Rings 184
5.5 Reduced Rank and Artinian Classical Ring of Fractions 191
5.6 Classical Krull Dimension 200
5.7 Krull Dimension 205
5.8 Relationship between Classical Krull Dimension and Krull Dimension 218
5.9 Notes and References 225
6. Rings with Finiteness Conditions 227
6.1 Some Noetherian Rings 227
6.2 Dedekind-finite Rings and Stably Finite Rings 232
6.3 FDI-Rings 249
6.4 Semiprime FDI-Rings 254
6.5 Notes and References 259
7. Modules with Change Property and Change Rings 260
7.1 The Exchange Property 261
7.2 The Azumaya Theorem 271
7.3 Cancellation Property 277
7.4 Exchange Rings 283
7.5 Notes and References 291
8. Hereditary and Semihereditary Rings. Piecewise Domains 294
8.1 Rickart Rings and Small’s Theorems 297
8.2 Dimensions of Hereditary and Semihereditary Rings 301
8.3 Right Hereditary and Right Semihereditary Prime Rings 305
8.4 Piecewise Domains. Right Hereditary Perfect Rings 313
8.5 Primely Triangular Rings. Structure of Piecewise Domains 316
8.6 Right Hereditary Triangular Rings 324
8.7 Noetherian Hereditary Primely Triangular Rings 326
8.8 Right Hereditary Species and Tensor Algebras 328
8.9 Notes and References 332
4. Homological Dimensions of Rings and Modules 119
4.1 Projective and Injective Dimension 120
4.2 Flat and Weak Dimension 122
4.3 Some Examples 124
4.4 Homological Characterization of Some Classes of Rings 130
4.5 Torsion-less Modules 136
4.6 Flat Modules and Coherent Rings 144
4.7 Modules over Formal Triangular Matrix Rings 152
4.8 Notes and References 158
5. Goldie and Krull Dimensions of Rings and Modules 160
5.1 Uniform Modules and Uniform Dimension 161
5.2 Injective Uniform Modules 171
5.3 Nonsingular Modules and Rings 175
5.4 Nonsingular Rings and Goldie Rings 184
5.5 Reduced Rank and Artinian Classical Ring of Fractions 191
5.6 Classical Krull Dimension 200
5.7 Krull Dimension 205
5.8 Relationship between Classical Krull Dimension and Krull Dimension 218
5.9 Notes and References 225
6. Rings with Finiteness Conditions 227
6.1 Some Noetherian Rings 227
6.2 Dedekind-finite Rings and Stably Finite Rings 232
6.3 FDI-Rings 249
6.4 Semiprime FDI-Rings 254
6.5 Notes and References 259
7. Modules with Change Property and Change Rings 260
7.1 The Exchange Property 261
7.2 The Azumaya Theorem 271
7.3 Cancellation Property 277
7.4 Exchange Rings 283
7.5 Notes and References 291
8. Hereditary and Semihereditary Rings. Piecewise Domains 294
8.1 Rickart Rings and Small’s Theorems 297
8.2 Dimensions of Hereditary and Semihereditary Rings 301
8.3 Right Hereditary and Right Semihereditary Prime Rings 305
8.4 Piecewise Domains. Right Hereditary Perfect Rings 313
8.5 Primely Triangular Rings. Structure of Piecewise Domains 316
8.6 Right Hereditary Triangular Rings 324
8.7 Noetherian Hereditary Primely Triangular Rings 326
8.8 Right Hereditary Species and Tensor Algebras 328
8.9 Notes and References 332
9. Serial Nonsingular Rings. Jacobsons’s Conjecture 334
9.1 Structure of Serial Right Noetherian Piecewise Domains 335
9.2 Structure of Serial Nonsingular Rings 339
9.3 Serial Rings with Noetherian Diagonal 346
9.4 The Krull Intersection Theorem 349
9.5 Jacobson’s Conjecture 352
9.6 Notes and References 357
References 359
