1 Free Modules,Projective,and Injective Modules 1
1．Free Modules 2
1A．Invariant Basis Number(IBN) 2
1B．Stable Finiteness 5
1C．The Rank Condition 9
1D．The Strong Rank Condition 12
1E．Synopsis 16
Exercises for §1 17
2．Projective Modules 21
2A．Basic Definitions and Examples 21
2B．Dual Basis Lemma and Invertible Modules 23
2C．Invertible Fractional Ideals 30
2D．The Picard Group of a Commutative Ring 34
2E．Hereditary and Semihereditary Rings 42
2F．Chase Small Examples 45
2G．Hereditary Artinian Rings 48
2H．Trace Ideals 51
Exercises for §2 54
3．Injective Modules 60
3A．Baer's Test for Iniectivity 60
3B．Self-Injective Rings 64
3C．Injectivity versus Divisibility 69
3D．Essential Extensions and Iniective Hulls 74
3E．Injectives over Right Noetherian Rings 80
3F．Indecomposable Injectives and Uniform Modules 83
3G．Iniectives over Some Artinian Rings 90
3H．Simple Injectives 96
31．Matlis' Theory 99
3J．Some Computations of Injective Hulls 105
3K．Applications to Chain Conditions 110
Exercises for §3 113
2 Flat Modules and Homological Dimensions 121
4．Flat and Faithfully Flat Modules 122
4A．Basic Properties and Flatness Tests 122
4B．Flatness,Torsion-Freeness,and von Neumann Regularity 127
4C．More Flatness Tests 129
4D．Finitely Presented(f.p.)Modules 131
4E．Finitely Generated Flat Modules 135
4F．Direct Products of Flat Modules 136
4G．Coherent Modules and Coherent Rings 140
4H．Semihereditary Rings Revisited 144
41．Faithfully Flat Modules 147
4J．Pure Exact Sequences 153
Exercises for §4 159
5．Homological Dimensions 165
5A．Schanuel's Lemma and Projective Dimensions 165
5B．Change of Rings 173
5C．Injective Dimensions 177
5D．Weak Dimensions of Rings 182
5E．Global Dimensions of Semiprimary Rings 187
5F．Global Dimensions of Local Rings 192
5G．Global Dimensions of Commutative Noetherian Rings 198
Exercises for §5 201
3 More Theory of Modules 207
6．Uniform Dimensions,Complements,and CS Modules 208
6A．Basic Definitions and Properties 208
6B．Complements and Closed Submodules 214
6C．Exact Sequences and Essential Closures 219
6D．CS Modules:Two Applications 221
6E．Finiteness Conditions on Rings 228
6F．Change of Rings 232
6G．Quasi-lnjective Modules 236
Exercises for §6 241
7．Singular Submodules and Nonsingular Rings 246
7A．Basic Definitions and Examples 246
7B．Nilpotency of the Right Singular Ideal 252
7C．Goldie Closures and the Reduced Rank 253
7D．Baer Rings and Rickart Rings 260
7E．Applications to Hereditary and Semihereditary Rings 265
Exercises for §7 268
8．Dense Submodules and Rational Hulls 272
8A．Basic Definitions and Examples 272
8B．Rational Hull of a Module 275
8C．Right Kasch Rings 280
Exercises for §8 284
4 Rings of Quotients 287
9．Noncommutative Localization 288
9A．"The Good" 288
9B．"The Bad" 290
9C．"The Ugly" 294
9D．An Embedding Theorem of A.Robinson 297
Exercises for §9 298
10．Classical Rings of Quotients 299
10A．Ore Localizations 299
10B．Right Ore Rings and Domains 303
10C．Polynomial Rings and Power Series Rings 308
10D．Extensions and Contractions 314
Exercises for §10 317
11．Right Goldie Rings and Goldie's Theorems 320
11A．Examples of Right Orders 320
11B．Right Orders in Semisimple Rings 323
11C．Some Applications of Goldie's Theorems 331
11D．Semiprime Rings 334
11E．Nil Multiplicatively Closed Sets 339
Exercises for §11 342
12．Artinian Rings of Quotients 345
12A．Goldie's ρ-Rank 345
12B．Right Orders in Right Artinian Rings 347
12C．The Commutative Case 351
12D．Noetherian Rings Need Not Be Ore 354
Exercises for §12 355
5 More Rings of Quotients 357
13．Maximal Rings of Quotients 358
13A．Endomorphism Ring of a Quasi-Injective Module 358
13B．Construction of Qr max(R) 365
13C．Another Description of Qr max(R) 369
13D．Theorems of Johnson and Gabriel 374
Exercises for §13 380
14．Martindale Rings of Quotients 383
14A．Semiprime Rings Revisited 383
14B．The Rings Qr(R)and Qs(R) 384
14C．The Extende4d Centroid 389
14D．Characterizations of Qr(R)and Qs(R) 392
14E．X-Inner Automorphisms 394
14F．A Matrix Ring Example 401
Exercises for §14 403
6 Frobenius and Quasi-Frobenius Rings 407
15．Quasi-Frobenius Rings 408
15A．Basic Definitions of QF Rings 408
15B．Projectives and Injectives 412
15C．Duality Properties 414
15D．Commutative QF Rings,and Examples 417
Exercises for §15 420
16．Frobenius Rings and Symmetric Algebras 422
16A．The Nakayama Permutation 422
16B．Definition of a Frobenius Ring 427
16C．Frobenius Algebras and QF Algebras 431
16D．Dimension Characterizations of Frobenius Algebras 434
16E．The Nakayama Automorphism 438
16F．Symmetric Algebras 441
16G．Why Frobenius? 450
Exercises for §16 453
7 Matrix Rings,Categories of Modules,and Morita Theory 459
17．Matrix Rings 461
17A．Characterizations and Examples 461
17B．First Instance of Module Category Equivalences 470
17C．Uniqueness of the Coefficient Ring 473
Exercises for §17 478
18．Morita Theory of Category Equivalences 480
18A．Categorical Properties 480
18B．Generators and Progenerators 483
18C．The Morita Context 485
18D．Morita Ⅰ，Ⅱ，Ⅲ 488
18E．Consequences of the Morita Theorems 490
18F．The Category σ［M］ 496
Exercises for §18 501
19．Morita Duality Theory 505
19A．Finite Cogeneration and Cogenerators 505
19B．Cogenerator Rings 510
19C．Classical Examples of Dualities 515
19D．Morita Dualities:Morita Ⅰ 518
19E．Consequences of Morita Ⅰ 522
19F．Linear Compactness and Reflexivity 527
19G．Morita Dualities:Morita Ⅱ 534
Exercises for §19 537
References 543
Name Index 549
Subject Index 553