模与环讲义

Preface

Notes to the Reader

Partial List of Notations

Partial List of Abbreviations

1 Free Modules， Projective， and Injective Modules

1. Free Modules

1A. Invariant Basis Number （IBN）

1B. Stable Finiteness

1C. The Rank Condition

1D. The Strong Rank Condition

1E. Synopsis

Exercises for 1

2. Projective Modules

2A. Basic Definitions and Examples

2B. Dual Basis Lemma and Inveriible Modules

2C. Invertible Fraaional Ideals

2D. The Picard Group of a Commutative Ring

2E. Hereditary and Semihereditary Rings

2F. Chase Small Examples

2G. Hereditary Artinian Rings

2H. Trace Ideals

Exercises for 2

3. Injective Modules

3A. Baer's Test for Injectivitv

3B. Self-Iniective Rings

3C. Injectivity versus Divisibility

3D. Essential Extensions and Injective Hulls

3E. Injectives over Right Noetherian Rings

3F. Indecomposable Injectives and Uniform Modules

3G. Injectives over Some Artinian Rings

3H. Simple Injcctives

3I. Matlis' Theory

3J. Some Computations oflnjective Hulls

3K. Applications to Chain Conditions

Exercises for 3

2 Flat Modules and Homological Dimensions

4. Flat and Faithfully Flat Modules

4A. Basic Properties and Flatness Tests

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

4C. More Flatness Tests

4D. Finitely Presented （f.p.） Modules

4E. Finitely Generated Flat Modules

4F. Direct Products of Flat Modules

4G. Coherent Modules and Coherent Rings

4H. Semihereditary Rings Revisited

4I. Faithfully Flat Modules

4J. Pure Exact Sequences

Exercises for 4

5. Homological Dimensions

5A. Schanuel's Lemma and Projective Dimensions

5B. Change of Rings

5C. Injectivc Dimensions

5D. Weak Dimensions of Rings

5E. Global Dimensions of Semiprimary Rings

5F. Global Dimensions of Local Rings

5G. Global Dimensions ofCommutative Noetherian Rings

Exercises for 5

3 More Theory of Modules

4 Rings of Quotients

5 More Rings of Quotients

6 Frobenius and Quasi-Frobenius Rings

7 Matrix Rings， Categories of Modules， and Morita Theory

References

Name Index

Subject Index

《模与环讲义》主要内容包括：Free Modules， Projective， and Injective Modules、Flat Modules and Homological Dimensions、More Theory of Modules、Rings of Quotients、More Rings of Quotients、Frobenius and Quasi-Frobenius Rings、Matrix Rings， Categories of Modules， and Morita Theory等七大章。