Commutative group rings of finite representation type
type: Item
instance of: scholarly article
title: Commutative group rings of finite representation type
zbMATH Open document ID: 0783.20005
DOI: 10.1016/0022-4049(92)90076-R
author: Lee Klingler
published in: Journal of Pure and Applied Algebra
publication date: 1993-01-16T00:00:00Z
review text: Let \(p\) be a rational prime and let \(\mathbb{Z}_ p\) be the localisation of \(\mathbb{Z}\) at \(p\). Let \(G\) be a finite abelian group of finite representation type, i.e. \(G\) has at most finitely many isomorphism classes of indecomposable \(\mathbb{Z}_ p G\)-lattices; the \(p\)-Sylow subgroup of \(G\) is necessarily cyclic of order at most \(p^ 2\). This paper determines the possible ranks of such \(\mathbb{Z}_ p G\)-lattices. It is shown that an indecomposable \(\mathbb{Z}_ p G\)-lattice may be embedded as a sublattice of \(\mathbb{Z}_ p G^{(4)}\) but that, if the commutativity of \(G\) is abandoned, the corresponding conclusion, even in a weaker sense, is false, thereby contradicting the author's own previous remark [J. Algebra 129, 26-74 (1990; Zbl 0688.20003)].
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 20C11
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16H05
zbMATH DE Number: 89580
zbMATH Keywords: finite abelian group
zbMATH Keywords: finite representation type
zbMATH Keywords: indecomposable \(\mathbb{Z}_ p G\)-lattices
zbMATH Keywords: ranks
MaRDI profile type: Publication
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full work available at URL: 0022-4049(92)90076-r
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description: scientific article; zbMATH DE number 89580