Endomorphism Rings of Abelian Groups
Gebonden Engels 2003 2003e druk 9781402014383Samenvatting
Every Abelian group can be related to an associative ring with an identity element, the ring of all its endomorphisms. Recently the theory of endomor phism rings of Abelian groups has become a rapidly developing area of algebra. On the one hand, it can be considered as a part of the theory of Abelian groups; on the other hand, the theory can be considered as a branch of the theory of endomorphism rings of modules and the representation theory of rings. There are several reasons for studying endomorphism rings of Abelian groups: first, it makes it possible to acquire additional information about Abelian groups themselves, to introduce new concepts and methods, and to find new interesting classes of groups; second, it stimulates further develop ment of the theory of modules and their endomorphism rings. The theory of endomorphism rings can also be useful for studies of the structure of additive groups of rings, E-modules, and homological properties of Abelian groups. The books of Baer [52] and Kaplansky [245] have played an important role in the early development of the theory of endomorphism rings of Abelian groups and modules. Endomorphism rings of Abelian groups are much stu died in monographs of Fuchs [170], [172], and [173]. Endomorphism rings are also studied in the works of Kurosh [287], Arnold [31], and Benabdallah [63].
Specificaties
Lezersrecensies
Inhoudsopgave
I: General Results on Endomorphism Rings. 1. Rings, Modules and Categories. 2. Abelian Groups. 3. Examples and Some Properties of Endomorphism Rings. 4. Torsion-Free Rings of Finite Rank. 5. Quasi-Endomorphism Rings of Torsion-Free Groups. 6. E-Modules and E-Rings. 7. Torsion-Free Groups Coinciding with Their Pseudo-Socles. 8. Irreducible Torsion-Free Groups.
II: Groups as Modules over Their Endomorphism Rings. 9. Endo-Artinian and Endo-Noetherian Groups. 10. Endo-Flat Primary Groups. 11. Endo-Finite Torsion-Free Groups of Finite Rank. 12. Endo-Projective and Endo-Generator Torsion-Free Groups of Finite Rank. 13. Endo-Flat Torsion-Free Groups of Finite Rank.
III: Ring Properties of Endomorphism Rings. 14. The Finite Topology. 15. Endomorphism Rings with the Minimum Condition. 16. Hom(A, B) as a Noethian Module over End(£Ii£). 18. Regular Endomorphism Rings. 19. Commutative and Local Endomorphism Rings.
IV: The Jacobson Radical of the Endomorphism Ring. 20. The Case of p-groups. 21. The Radical of the Endomorphism Ring of a Torsion-Free Group of Finite Rank. 22. The Radical of the Endomorphism Ring of Algebraically Compact and Completely Decomposable Torsion-Free Groups. 23. The Nilpotence of the Radicals N (End(G)) and J (End(G)).
V: Isomorphism and Realization Theorems. 24. The Baer Kaplansky Theorem. 25. Continuous and Discrete Isomorphisms of Endomorphism Rings. 26. Endomorphism Rings of Groups with Large Divisible Subgroups. 27. Endomorphism Rings of Mixed Groups of Torsion-Free Rank 1. 28. The Corner Theorem on Split Realization. 29. Realizations for Endomorphism Rings of Torsion-Free Groups. 30. The Realization Problem for Endomorphism Rings of Mixed Groups.
VI: Hereditary Endomorphism Rings. 31.Self-Small Groups. 32. Categories of Groups andModules over Endomorphism Rings. 33. Faithful Groups. 34. Faithful Endo-Flat Groups. 35. Groups with Right Hereditary Endomorphism Rings. 36. Groups of Generalized Rank 1. 37. Torsion-Free Groups with Hereditary Endomorphism Rings. 38. Maximal Orders as Endomorphism Rings. 39. p-Semisimple Groups.
VII: Fully Transitive Groups. 40. Homogeneous Fully Transitive Groups. 41. Groups whose Quasi-Endomorphism Rings are Division Rings. 42. Fully Transitive Groups Coinciding with Their Pseudo-Socles. 43. Fully Transitive Groups with Restrictions on Element Types. 44. Torsion-Free Groups of p-Ranks = 1.
References. Index.
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