New results related to a conjecture of Moore.
type: Item
instance of: scholarly article
title: New results related to a conjecture of Moore.
zbMATH Open document ID: 1278.16027
published in: Archiv der Mathematik
publication date: 2013-04-09T00:00:00Z
review text: Let \(\Gamma'\) be a subgroup of a group \(\Gamma\) and let \(R\) be a ring with identity. We say that the triple \((\Gamma,\Gamma',R)\) satisfies Moore's condition if \(\Gamma'\) is of finite index in \(\Gamma\) and for all \(x\in(\Gamma-\Gamma')\) either there is an integer \(n\) such that \(1\neq x^n\in\Gamma'\) or the order of \(x\) is invertible in \(R\). Let \(R\Gamma\) be the group ring of the group \(\Gamma\) over the ring \(R\) and let \(M\) be an arbitrary \(R\Gamma\)-module, whose restriction to \(R\Gamma'\) is projective. If the triple \((\Gamma,\Gamma',R)\) satisfies Moore's condition, then the Moore conjecture asserts that \(M\) is also a projective \(R\Gamma\)-module [see \textit{L. G. Chouinard}, J. Pure Appl. Algebra 7, 287-302 (1976; Zbl 0327.20020)]. In this paper the authors consider a similar analogue of this conjecture on injective modules. They show that the conjecture on injective modules is true for a very large class of groups. Moreover, they prove that the validity of the conjecture on injective modules implies the validity of it on projective and on flat modules. In addition, suppose that \(R\) is a commutative ring and \(M\) is an \(R\Gamma\)-module. Then it is proved that \(M\) is Gorenstein projective (resp. injective) if and only if \(M\) is Gorenstein projective (resp. injective) as an \(R\Gamma'\)-module [see \textit{E. E. Enochs} and \textit{O. M. G. Jenda}, Math. Z. 220, No. 4, 611-633 (1995; Zbl 0845.16005)].
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16D50
zbMATH DE Number: 6152308
zbMATH Keywords: Gorenstein projective modules
zbMATH Keywords: Moore conjecture
zbMATH Keywords: injectivity over group rings
zbMATH Keywords: Kropholler hierarchy
zbMATH Keywords: injective left modules
zbMATH Keywords: Gorenstein injective modules
zbMATH Keywords: subgroups of finite index
zbMATH Keywords: flat modules
Wikidata QID: Q123200437
author: Abdolnaser Bahlekeh
author: Shokrollah Salarian
reviewed by: S. V. Mihovski
MaRDI profile type: Publication
full work available at URL: s00013-013-0490-7
OpenAlex ID: W1963955798
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DOI: 10.1007/S00013-013-0490-7
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description: scientific article; zbMATH DE number 6152308