Ebook: Artinian Modules over Group Rings

From the reviews:

“The theory of modules over group rings RG for infinite groups G over arbitrary rings R is a very extensive and complex field of research with a great number of scattered results. … Since many of the results appear for the first time in a book it can be recommended warmly to any expert in this field, but also for graduate students who are presented the beauty of the interplay of the theories of groups, rings and representations.” (G. Kowol, Monatshefte für Mathematik, Vol. 152 (4), December, 2007)

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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.

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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.

Content:
Front Matter....Pages i-xii
Modules with chain conditions....Pages 1-11
Ranks of groups....Pages 13-23
Some generalized nilpotent groups....Pages 25-35
Artinian modules and the socle....Pages 37-44
Reduction to subgroups of finite index....Pages 45-52
Modules over Dedekind domains....Pages 53-62
The Kovacs-Newman theorem....Pages 63-80
Hartley’s classes of modules....Pages 81-91
The injectivity of some simple modules....Pages 93-110
Direct decompositions in artinian modules....Pages 111-130
On the countability of artinian modules over FC-hypercentral groups....Pages 131-152
Artinian modules over periodic abelian groups....Pages 153-160
Nearly injective modules....Pages 161-166
Artinian modules over abelian groups of finite section rank....Pages 167-177
The injective envelopes of simple modules over group rings....Pages 179-188
Quasifinite modules....Pages 189-210
Some applications: splitting over the locally nilpotent residual....Pages 211-225
Back Matter....Pages 227-247

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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.

Content:
Front Matter....Pages i-xii
Modules with chain conditions....Pages 1-11
Ranks of groups....Pages 13-23
Some generalized nilpotent groups....Pages 25-35
Artinian modules and the socle....Pages 37-44
Reduction to subgroups of finite index....Pages 45-52
Modules over Dedekind domains....Pages 53-62
The Kovacs-Newman theorem....Pages 63-80
Hartley’s classes of modules....Pages 81-91
The injectivity of some simple modules....Pages 93-110
Direct decompositions in artinian modules....Pages 111-130
On the countability of artinian modules over FC-hypercentral groups....Pages 131-152
Artinian modules over periodic abelian groups....Pages 153-160
Nearly injective modules....Pages 161-166
Artinian modules over abelian groups of finite section rank....Pages 167-177
The injective envelopes of simple modules over group rings....Pages 179-188
Quasifinite modules....Pages 189-210
Some applications: splitting over the locally nilpotent residual....Pages 211-225
Back Matter....Pages 227-247
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