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This book is intended to provide a reasonable selr-contained account of a major portion of the general theory of rings and modules suitable as a text for introductory and more advanced graduate courses. We assume the familiarity with rings usually acquired in standard undergraduate algebra courses. Our general approach is categorical rather than arithmetical. The continuting theme of the text is the study of the relationship between the one-sided ideal structure that a ring may possess and the behavior of its categories of modules. 本書為英文版。
目次
Preface O. Preliminaries Chapter 1: Rinos， Modules and Homomorphisms 1. Review of Rings and their Homomorphisms 2. Modules and Submodules 3. Homomorphisms of Modules 4. Categories of Modules； Endomorphism Rings Chapter 2: Direct Sums and Products 5. Direct Summands 6. Direct Sums and Products of Modules 7. Decomposition of Rings 8. Generating and Cogenerating Chapter 3: Finiteness Conditions for Modules 9. Semisimple Modules--The Socle and the Radical 10. Finitely Generated and Finitely Cogenerated Modules-Chain Conditions 11. Modules with Composition Series 12. Indecomposable Decompositions of Modules Chapter 4: Classical Ring-Structure Theorems 13. Semisimple Rings 14. The Density Theorem 15. The Radical of a Ring-Local Rings and Artinian Rings Chapter 5: Functors Between Module Categories 16.The Hom Functors and Exactness-Projectivity and Injectivity 17.Projective Modules and Generators 18.Injective Modules and Cogenerators 19.The Tensor Functors and Flat Modules 20.Natural Transformations Chapter 6: Equivalence and Duality for Module Categories 21.Equivalent Rings 22.The Morita Characterizations of Equivalence 23.Dualities 24.Morita Dualities Chapter 7: Injective Modules,Projective Modules,and Their Decompositions 25.Injective Modules and Noetherian Rings-The Faith-Walker Theorems 26.Direct Sums of Countably Generated Modules-With Local Endomorphism Rings 27.Semiperfect Rings 28.Perfect Rings 29.Modules with Perfect Endomorphism Rings Chapter 8: Classical Artinian Rings 30.Artinian Rings with Duality 31.Injective Projective Modules 32.Serial Rings Bibliography Index
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