Modules over Endomorphism Rings

An extensive synthesis of recent work in the study of endomorphism rings and their modules.

Theodore G. Faticoni (Author)

9780521199605, Cambridge University Press

Hardback, published 22 October 2009

392 pages, 210 exercises
24 x 16 x 2.5 cm, 0.73 kg

'Theodore Faticoni enriches the mathematical literature with a comprehensive and up-to-date account of the study of a large class of abelian groups presenting all the relevant results in a very well organised fashion … the book is a succession of very sharp results, which are the fruits of methodical and thorough investigations started about 40 years ago by the author. Page after page, Faticoni engages the reader in a friendly mathematical discussion. As incentive, he provides some historical motivations for the questions he considers, for example, the classification of algebraic number fields, which goes back to Gauss and Kummer, while numerous examples, exercises and problems for future research at the end of each chapter will certainly provide the reader with some stimulating challenges.' Bulletin of the London Mathematical Society

This is an extensive synthesis of recent work in the study of endomorphism rings and their modules, bringing together direct sum decompositions of modules, the class number of an algebraic number field, point set topological spaces, and classical noncommutative localization. The main idea behind the book is to study modules G over a ring R via their endomorphism ring EndR(G). The author discusses a wealth of results that classify G and EndR(G) via numerous properties, and in particular results from point set topology are used to provide a complete characterization of the direct sum decomposition properties of G. For graduate students this is a useful introduction, while the more experienced mathematician will discover that the book contains results that are not otherwise available. Each chapter contains a list of exercises and problems for future research, which provide a springboard for students entering modern professional mathematics.

Preface
1. Preliminary results
2. Class number of an Abelian group
3. Mayer-Vietoris sequences
4. Lifting units
5. The conductor
6. Conductors and groups
7. Invertible fractional ideals
8. L-Groups
9. Modules and homotopy classes
10
Tensor functor equivalences
11. Characterizing endomorphisms
12. Projective modules
13. Finitely generated modules
14. Rtffr E-projective groups
15. Injective endomorphism modules
16. A diagram of categories
17. Diagrams of Abelian groups
18. Marginal isomorphisms
References
Index.

Subject Areas: Algebra [PBF]