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AuthorAnderson, Frank W. author
TitleRings and Categories of Modules [electronic resource] / by Frank W. Anderson, Kent R. Fuller
ImprintNew York, NY : Springer New York : Imprint: Springer, 1974
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Descript IX, 339 p. online resource


This book is intended to provide a reasonably self-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 familยญ iarity with rings usually acquired in standard undergraduate algebra courses. Our general approach is categorical rather than arithmetical. The continuing 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. Following a brief outline of set-theoretic and categorical foundations, the text begins with the basic definitions and properties of rings, modules and homomorphisms and ranges through comprehensive treatments of direct sums, finiteness conditions, the Wedderburn-Art in Theorem, the Jacobson radical, the hom and tensor functions, Morita equivalence and duality, deยญ composition theory of injective and projective modules, and semiperfect and perfect rings. Both to illustrate the text and to extend it we have included a substantial number of exercises covering a wide spectrum of difficulty. There are, of course, many important areas of ring and module theory that the text does not touch upon. For example, we have made no attempt to cover such subjects as homology, rings of quotients, or commutative ring theory


ยง0. Preliminaries -- 1: Rings, Modules and Homomorphisms -- ยง1. Review of Rings and Their Homomorphisms -- ยง2. Modules and Submodules -- ยง3. Homomorphisms of Modules -- ยง4. Categories of Modules; Endomorphism Rings -- 2: Direct Sums and Products -- ยง5. Direct Summands -- ยง6. Direct Sums and Products of Modules -- ยง7. Decomposition of Rings -- ยง8. Generating and Cogenerating -- 3: Finiteness Conditions for Modules -- ยง9. Semisimple Modulesโ{128}{148}The Socle and the Radical -- ยง10. Finitely Generated and Finitely Cogenerated Modulesโ{128}{148} Chain Conditions -- ยง11. Modules With Composition Series -- ยง12. Indecomposable Decompositions of Modules -- 4: Classical Ring-Structure Theorems -- ยง13. Semisimple Rings -- ยง14. The Density Theorem -- ยง15. The Radical of a Ringโ{128}{148}Local Rings and Artinian Rings -- 5: Functions Between Module Categories -- ยง16. The Hom Functors and Exactness โ{128}{148} Projectivity and Injectivity -- ยง17. Projective Modules and Generators -- ยง18. Injective Modules and Cogenerators -- ยง19. The Tensor Functors and Flat Modules -- ยง20. Natural Transformations -- 6: Equivalence and Duality for Module Categories -- ยง21. Equivalent Rings -- ยง22. The Morita Characterizations of Equivalence -- ยง23. Dualities -- ยง24. Morita Dualities -- 7: Injective Modules, Projective Modules and Their Decompositions -- ยง25. Injective Modules and Noetherian Rings-The Faith-Walker Theorems -- ยง26. Direct Sums of Countably Generated Modules-With Local Endomorphism Rings -- ยง27. Semiperfect Rings -- ยง28. Perfect Rings -- ยง29. Modules With Perfect Endomorphism Rings

Mathematics Algebra Physics Mathematics Algebra Physics general


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