Artinian Modules over Group Rings

by Kurdachenko, Leonid A.; Otal, Javier; Subbotin, Igor Ya

• ISBN13: 9783764377649
• ISBN10: 376437764X
• Format: Paperback
• Copyright: 2006-12-04
• Publisher: Birkhauser

Summary

This book highlights important developments on artinian modules over group rings of generalized nilpotent groups. Along with traditional topics such as direct decompositions of artinian modules, criteria of complementability for some important modules, and criteria of semisimplicity of artinian modules, it also focuses on recent advanced results on these matters. The theory of modules over groups has its own specific character that plays an imperative role here and, for example, allows a significant generalization of the classical Maschke Theorem on some classes of infinite groups. Conversely, it leads to establishing direct decompositions of artinian modules related to important natural formations, which, in turn, find very efficient applications in infinite groups. As self-contained as possible, this book will be useful for students as well as for experts in group theory, ring theory, and module theory.

Table of Contents

Preface	p. ix
Modules with chain conditions	p. 1
Ranks of groups	p. 13
Some generalized nilpotent groups	p. 25
Artinian modules and the socle	p. 37
Reduction to subgroups of finite index	p. 45
Modules over Dedekind domains	p. 53
The Kovacs-Newman theorem	p. 63
Hartley's classes of modules	p. 81
The injectivity of some simple modules	p. 93
Direct decompositions in artinian modules	p. 111
On the countability of artinian modules over FC-hypercentral groups	p. 131
Artinian modules over periodic abelian groups	p. 153
Nearly injective modules	p. 161
Artinian modules over abelian groups of finite section rank	p. 167
The injective envelopes of simple modules over group rings	p. 179
Quasifinite modules	p. 189
Some applications: splitting over the locally nilpotent residual	p. 211
Bibliography	p. 227
Index	p. 245
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