Rings and Their Modules (eBook)

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Paul E. Bland, Eastern Kentucky University, USA.

Preface 6
About the Text 6
Acknowledgements 9
Contents 10
0 Preliminaries 16
0.1 Classes, Sets and Functions 16
Partial Orders and Equivalence Relations 17
Zorn’s Lemma and Well-Ordering 18
0.2 Ordinal and Cardinal Numbers 19
0.3 Commutative Diagrams 19
0.4 Notation and Terminology 19
Problem Set 20
1 Basic Properties of Rings and Modules 22
1.1 Rings 22
Problem Set 1.1 27
1.2 Left and Right Ideals 29
Factor Rings 32
Problem Set 1.2 32
1.3 Ring Homomorphisms 36
Problem Set 1.3 39
1.4 Modules 40
Factor Modules 44
Problem Set 1.4 45
1.5 Module Homomorphisms 47
Problem Set 1.5 51
2 Fundamental Constructions 54
2.1 Direct Products and Direct Sums 54
Direct Products 54
External Direct Sums 58
Internal Direct Sums 60
Problem Set 2.1 64
2.2 Free Modules 66
Rings with Invariant Basis Number 71
Problem Set 2.2 75
2.3 Tensor Products of Modules 78
Problem Set 2.3 84
3 Categories 86
3.1 Categories 86
Functors 90
Properties of Morphisms 92
Problem Set 3.1 94
3.2 Exact Sequences in ModR 97
Split Short Exact Sequences 99
Problem Set 3.2 101
3.3 Hom and < 8>
Properties of Hom 105
Properties of Tensor Products 109
Problem Set 3.3 110
3.4 Equivalent Categories and Adjoint Functors 112
Adjoints 114
Problem Set 3.4 117
4 Chain Conditions 119
4.1 Generating and Cogenerating Classes 119
Problem Set 4.1 121
4.2 Noetherian and Artinian Modules 122
Problem Set 4.2 133
4.3 Modules over Principal Ideal Domains 135
Free Modules over a PID 139
Finitely Generated Modules over a PID 143
Problem Set 4.3 149
5 Injective, Projective, and Flat Modules 150
5.1 Injective Modules 150
Injective Modules and the Functor HomR(—,M) 155
Problem Set 5.1 157
5.2 Projective Modules 159
Projective Modules and the Functor HomR (M, —) 163
Hereditary Rings 163
Semihereditary Rings 166
Problem Set 5.2 167
5.3 Flat Modules 169
Flat Modules and the Functor M < g>
Coherent Rings 174
Regular Rings and Flat Modules 178
Problem Set 5.3 180
5.4 Quasi-Injective and Quasi-Projective Modules 184
Problem Set 5.4 185
6 Classical Ring Theory 186
6.1 The Jacobson Radical 186
Problem Set 6.1 192
6.2 The Prime Radical 193
Prime Rings 194
Semiprime Rings 199
Problem Set 6.2 201
6.3 Radicals and Chain Conditions 203
Problem Set 6.3 205
6.4 Wedderburn–Artin Theory 206
Problem Set 6.4 218
6.5 Primitive Rings and Density 220
Problem Set 6.5 225
6.6 Rings that Are Semisimple 226
Problem Set 6.6 230
7 Envelopes and Covers 231
7.1 Injective Envelopes 231
Problem Set 7.1 234
7.2 Projective Covers 236
The Radical of a Projective Module 238
Semiperfect Rings 242
Perfect Rings 252
Problem Set 7.2 256
7.3 QI-Envelopes and QP-Covers 260
Quasi-Injective Envelopes 260
Quasi-Projective Covers 263
Problem Set 7.3 267
8 Rings and Modules of Quotients 269
8.1 Rings of Quotients 269
The Noncommutative Case 269
The Commutative Case 276
Problem Set 8.1 277
8.2 Modules of Quotients 279
Problem Set 8.2 282
8.3 Goldie’s Theorem 285
Problem Set 8.3 291
8.4 The Maximal Ring of Quotients 292
Problem Set 8.4 306
9 Graded Rings and Modules 309
9.1 Graded Rings and Modules 309
Graded Rings 309
Graded Modules 314
Problem Set 9.1 320
9.2 Fundamental Concepts 323
Graded Direct Products and Sums 324
Graded Tensor Products 326
Graded Free Modules 327
Problem Set 9.2 328
9.3 Graded Projective, Graded Injective and Graded Flat Modules 329
Graded Projective and Graded Injective Modules 329
Graded Flat Modules 333
Problem Set 9.3 334
9.4 Graded Modules with Chain Conditions 335
Graded Noetherian and Graded Artinian Modules 335
Problem Set 9.4 340
9.5 More on Graded Rings 340
The Graded Jacobson Radical 340
Graded Wedderburn–Artin Theory 342
Problem Set 9.5 344
10 More on Rings and Modules 346
10.1 Reflexivity and Vector Spaces 347
Problem Set 10.1 349
10.2 Reflexivity and R-modules 350
Self-injective Rings 351
Kasch Rings and Injective Cogenerators 354
Semiprimary Rings 356
Quasi-Frobenius Rings 357
Problem Set 10.2 361
11 Introduction to Homological Algebra 363
11.1 Chain and Cochain Complexes 363
Homology and Cohomology Sequences 368
Problem Set 11.1 372
11.2 Projective and Injective Resolutions 374
Problem Set 11.2 380
11.3 Derived Functors 382
Problem Set 11.3 387
11.4 Extension Functors 388
Right Derived Functors of HomR(—, X) 388
Right Derived Functors of HomR(X, —) 396
Problem Set 11.4 401
11.5 Torsion Functors 405
Left Derived Functors of — < S>
Problem Set 11.5 409
12 Homological Methods 410
12.1 Projective and Injective Dimension 410
Problem Set 12.1 417
12.2 Flat Dimension 418
Problem Set 12.2 422
12.3 Dimension of Polynomial Rings 424
Problem Set 12.3 432
12.4 Dimension of Matrix Rings 432
Problem Set 12.4 436
12.5 Quasi-Frobenius Rings Revisited 436
More on Reflexive Modules 436
Problem Set 12.5 443
A Ordinal and Cardinal Numbers 444
Ordinal Numbers 444
Cardinal Numbers 448
Problem Set 450
Bibliography 452
List of Symbols 456
Index 458

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"This recent book is a welcome addition to the textbook literature on the subject of (mostly noncommutative) ring theory [...] The author of this text is clearly interested in making this material, and what follows, as accessible as possible to a graduate student. [...] This book has all the attributes of an excellent text. The writing is clear and reader-friendly, and there are both good examples and a reasonable number of exercises of varying difficulty. ... It is highly recommended, both as a text for ring theory courses at the various levels described above, or as an accessible resource for graduate students with an interest in pursuing research in ring theory."
Mark Hunacek, MAA Reviews