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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