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**Artinian modules over group rings.**
*(English)*
Zbl 1110.16001

Frontiers in Mathematics. Basel: Birkhäuser (ISBN 3-7643-7764-X/pbk). xii, 247 p. (2007).

The theory of group rings not only gives a nice source of examples of noncommutative as well as commutative rings, but also provides the basic connections between ring theory and the theory of group representations. On the other hand, ideas and results in ring theory have important impact on the development of the theory of representations of groups. Thus arise fundamental connections between the theory of groups, group representations, group rings and modules over group rings.

This book highlights important developments on Artinian modules over group rings of generalized nilpotent groups. Along with traditional topics it also focuses on recent advanced results on these matters. In this direction we indicate the chapters Modules with chain condition, Artinian modules and the socle, Modules over Dedekind domains, The injectivity of some simple modules, Nearly injective modules, and Quasifinite modules. To the theory of groups the authors devote two chapters: Rank of groups, and Some generalized nilpotent groups.

The theory of modules over group rings has its own specific character that plays an imperative role here and, for example, allows a significant generalization of Maschke’s classical Theorem to some classes of infinite groups. Such problems are discussed in chapter 7, The Kovacs-Newman theorem. 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. In this direction we mark the chapters On the countability of Artinian modules over FC-hypercentral groups, Artinian modules over Abelian groups of finite section rank, and The injective envelopes of simple modules over group rings. The last chapter is dedicated to some group theoretical results about the splitting of a group over its locally nilpotent residual.

The authors focused their study on Artinian modules because they note that the Noetherian modules are presented well enough elsewhere. The bibliography includes 313 titles preferably of the last 50 years. 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.

This book highlights important developments on Artinian modules over group rings of generalized nilpotent groups. Along with traditional topics it also focuses on recent advanced results on these matters. In this direction we indicate the chapters Modules with chain condition, Artinian modules and the socle, Modules over Dedekind domains, The injectivity of some simple modules, Nearly injective modules, and Quasifinite modules. To the theory of groups the authors devote two chapters: Rank of groups, and Some generalized nilpotent groups.

The theory of modules over group rings has its own specific character that plays an imperative role here and, for example, allows a significant generalization of Maschke’s classical Theorem to some classes of infinite groups. Such problems are discussed in chapter 7, The Kovacs-Newman theorem. 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. In this direction we mark the chapters On the countability of Artinian modules over FC-hypercentral groups, Artinian modules over Abelian groups of finite section rank, and The injective envelopes of simple modules over group rings. The last chapter is dedicated to some group theoretical results about the splitting of a group over its locally nilpotent residual.

The authors focused their study on Artinian modules because they note that the Noetherian modules are presented well enough elsewhere. The bibliography includes 313 titles preferably of the last 50 years. 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.

Reviewer: S. V. Mihovski (Plovdiv)

### MSC:

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

16P20 | Artinian rings and modules (associative rings and algebras) |

16S34 | Group rings |

20C07 | Group rings of infinite groups and their modules (group-theoretic aspects) |

16P70 | Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras) |

20F19 | Generalizations of solvable and nilpotent groups |

20F24 | FC-groups and their generalizations |