Module Theory, Extending Modules and Generalizations

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Birkhäuser, May 13, 2016 - Mathematics - 369 pages

The main focus of this monograph is to offer a comprehensive presentation of known and new results on various generalizations of CS-modules and CS-rings. Extending (or CS) modules are generalizations of injective (and also semisimple or uniform) modules. While the theory of CS-modules is well documented in monographs and textbooks, results on generalized forms of the CS property as well as dual notions are far less present in the literature.

With their work the authors provide a solid background to module theory, accessible to anyone familiar with basic abstract algebra. The focus of the book is on direct sums of CS-modules and classes of modules related to CS-modules, such as relative (injective) ejective modules, (quasi) continuous modules, and lifting modules. In particular, matrix CS-rings are studied and clear proofs of fundamental decomposition results on CS-modules over commutative domains are given, thus complementing existing monographs in this area.

Open problems round out the work and establish the basis for further developments in the field. The main text is complemented by a wealth of examples and exercises.

 

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Contents

Chapter 1 Introducing Modules
1
Chapter 2 Types of Relative Injectivity
75
Chapter 3 Extending Property and Related Concepts
117
Chapter 4 Inner Generalizations of Extending Modules
175
Chapter 5 Outer Generalizations of Extending Modules
247
Chapter 6 Dual Goldie and ECcomplement Versions of the Extending Property
321
Chapter 7 Open Problems and Questions
347
Appendix Construction of Rings of Quotients via Gabriel Topologies
349
Bibliography
357
Index
367
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