On modules over group rings.
type: Item
instance of: scholarly article
title: On modules over group rings.
zbMATH Open document ID: 1307.16020
author: Tsiu-Kwen Lee
author: Yiqiang Zhou
author: M. Tamer Koşan
published in: Algebras and Representation Theory
publication date: 2014-06-23T00:00:00Z
review text: Let \(RG\) be the group ring of a group \(G\) over an associative ring \(R\) with identity. Suppose that \(M\) is a unitary right \(R\)-module and denote by \(MG\) the set of all formal finite sums of the form \(\sum_{g\in G}m_gg\), where \(m_g\in M\). In a natural way \(MG\) is a right \(RG\)-module, called the group module of \(G\) by \(M\) over \(RG\). The objective of the paper is to study basic properties of \(MG\) in terms of the properties of \(M\) and \(G\). In particular, the authors prove the module-theoretic versions of several well-known results of group rings, including Maschke's theorem, the classical characterizations of right self-injective group rings and of von Neumann regular group rings. For instance, it is proved that \(MG\) is a semisimple \(RG\)-module if and only if \(M\) is a semisimple \(R\)-module and \(G\) is a finite group whose order is invertible in \(\text{End}_R(M)\). The second main result asserts that \(MG\) is a regular \(RG\)-module if and only if \(M\) is a regular \(R\)-module, \(G\) is locally finite and the order of each finite subgroup of \(G\) is invertible in \(\text{End}_R(M)\). Moreover, \(MG\) is an injective \(RG\)-module if and only if \(M\) is an injective \(R\)-module and \(G\) is a finite group. The paper contains many other interesting results.
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16E50
zbMATH DE Number: 6306455
zbMATH Keywords: modules over group rings
zbMATH Keywords: group modules
zbMATH Keywords: Maschke theorem
zbMATH Keywords: semisimple modules
zbMATH Keywords: regular modules
zbMATH Keywords: injective modules
zbMATH Keywords: FP-injective modules
zbMATH Keywords: right self-injective group rings
zbMATH Keywords: von Neumann regular group rings
Wikidata QID: Q57787640
reviewed by: S. V. Mihovski
MaRDI profile type: Publication
full work available at URL: s10468-012-9388-5
OpenAlex ID: W2005280423
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DOI: 10.1007/S10468-012-9388-5
description: scientific article; zbMATH DE number 6306455