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1 Free Modules,Projective,and Injective Modules1

1．Free Modules2

1A．Invariant Basis Number(IBN)2

1B．Stable Finiteness5

1C．The Rank Condition9

1D．The Strong Rank Condition12

1E．Synopsis16

Exercises for §117

2．Projective Modules21

2A．Basic Definitions and Examples21

2B．Dual Basis Lemma and Invertible Modules23

2C．Invertible Fractional Ideals30

2D．The Picard Group of a Commutative Ring34

2E．Hereditary and Semihereditary Rings42

2F．Chase Small Examples45

2G．Hereditary Artinian Rings48

2H．Trace Ideals51

Exercises for §254

3．Injective Modules60

3A．Baer's Test for Iniectivity60

3B．Self-Injective Rings64

3C．Injectivity versus Divisibility69

3D．Essential Extensions and Iniective Hulls74

3E．Injectives over Right Noetherian Rings80

3F．Indecomposable Injectives and Uniform Modules83

3G．Iniectives over Some Artinian Rings90

3H．Simple Injectives96

31．Matlis' Theory99

3J．Some Computations of Injective Hulls105

3K．Applications to Chain Conditions110

Exercises for §3113

2 Flat Modules and Homological Dimensions121

4．Flat and Faithfully Flat Modules122

4A．Basic Properties and Flatness Tests122

4B．Flatness,Torsion-Freeness,and von Neumann Regularity127

4C．More Flatness Tests129

4D．Finitely Presented(f.p.)Modules131

4E．Finitely Generated Flat Modules135

4F．Direct Products of Flat Modules136

4G．Coherent Modules and Coherent Rings140

4H．Semihereditary Rings Revisited144

41．Faithfully Flat Modules147

4J．Pure Exact Sequences153

Exercises for §4159

5．Homological Dimensions165

5A．Schanuel's Lemma and Projective Dimensions165

5B．Change of Rings173

5C．Injective Dimensions177

5D．Weak Dimensions of Rings182

5E．Global Dimensions of Semiprimary Rings187

5F．Global Dimensions of Local Rings192

5G．Global Dimensions of Commutative Noetherian Rings198

Exercises for §5201

3 More Theory of Modules207

6．Uniform Dimensions,Complements,and CS Modules208

6A．Basic Definitions and Properties208

6B．Complements and Closed Submodules214

6C．Exact Sequences and Essential Closures219

6D．CS Modules:Two Applications221

6E．Finiteness Conditions on Rings228

6F．Change of Rings232

6G．Quasi-lnjective Modules236

Exercises for §6241

7．Singular Submodules and Nonsingular Rings246

7A．Basic Definitions and Examples246

7B．Nilpotency of the Right Singular Ideal252

7C．Goldie Closures and the Reduced Rank253

7D．Baer Rings and Rickart Rings260

7E．Applications to Hereditary and Semihereditary Rings265

Exercises for §7268

8．Dense Submodules and Rational Hulls272

8A．Basic Definitions and Examples272

8B．Rational Hull of a Module275

8C．Right Kasch Rings280

Exercises for §8284

4 Rings of Quotients287

9．Noncommutative Localization288

9A．"The Good"288

9B．"The Bad"290

9C．"The Ugly"294

9D．An Embedding Theorem of A.Robinson297

Exercises for §9298

10．Classical Rings of Quotients299

10A．Ore Localizations299

10B．Right Ore Rings and Domains303

10C．Polynomial Rings and Power Series Rings308

10D．Extensions and Contractions314

Exercises for §10317

11．Right Goldie Rings and Goldie's Theorems320

11A．Examples of Right Orders320

11B．Right Orders in Semisimple Rings323

11C．Some Applications of Goldie's Theorems331

11D．Semiprime Rings334

11E．Nil Multiplicatively Closed Sets339

Exercises for §11342

12．Artinian Rings of Quotients345

12A．Goldie's ρ-Rank345

12B．Right Orders in Right Artinian Rings347

12C．The Commutative Case351

12D．Noetherian Rings Need Not Be Ore354

Exercises for §12355

5 More Rings of Quotients357

13．Maximal Rings of Quotients358

13A．Endomorphism Ring of a Quasi-Injective Module358

13B．Construction of Qr max(R)365

13C．Another Description of Qr max(R)369

13D．Theorems of Johnson and Gabriel374

Exercises for §13380

14．Martindale Rings of Quotients383

14A．Semiprime Rings Revisited383

14B．The Rings Qr(R)and Qs(R)384

14C．The Extende4d Centroid389

14D．Characterizations of Qr(R)and Qs(R)392

14E．X-Inner Automorphisms394

14F．A Matrix Ring Example401

Exercises for §14403

6 Frobenius and Quasi-Frobenius Rings407

15．Quasi-Frobenius Rings408

15A．Basic Definitions of QF Rings408

15B．Projectives and Injectives412

15C．Duality Properties414

15D．Commutative QF Rings,and Examples417

Exercises for §15420

16．Frobenius Rings and Symmetric Algebras422

16A．The Nakayama Permutation422

16B．Definition of a Frobenius Ring427

16C．Frobenius Algebras and QF Algebras431

16D．Dimension Characterizations of Frobenius Algebras434

16E．The Nakayama Automorphism438

16F．Symmetric Algebras441

16G．Why Frobenius?450

Exercises for §16453

7 Matrix Rings,Categories of Modules,and Morita Theory459

17．Matrix Rings461

17A．Characterizations and Examples461

17B．First Instance of Module Category Equivalences470

17C．Uniqueness of the Coefficient Ring473

Exercises for §17478

18．Morita Theory of Category Equivalences480

18A．Categorical Properties480

18B．Generators and Progenerators483

18C．The Morita Context485

18D．Morita Ⅰ，Ⅱ，Ⅲ488

18E．Consequences of the Morita Theorems490

18F．The Category σ［M］496

Exercises for §18501

19．Morita Duality Theory505

19A．Finite Cogeneration and Cogenerators505

19B．Cogenerator Rings510

19C．Classical Examples of Dualities515

19D．Morita Dualities:Morita Ⅰ518

19E．Consequences of Morita Ⅰ522

19F．Linear Compactness and Reflexivity527

19G．Morita Dualities:Morita Ⅱ534

Exercises for §19537

References543

Name Index549

Subject Index553