简介
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.
本书为英文版。
目录
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 ora ring--local rings and artinian rings
chapter 5: functors between module categories
16. the horn 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
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 ora ring--local rings and artinian rings
chapter 5: functors between module categories
16. the horn 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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