Primitive localizations of group algebras of polycyclic-by-finite groups
type: Item
instance of: scholarly article
title: Primitive localizations of group algebras of polycyclic-by-finite groups
zbMATH Open document ID: 0932.16032
author: P. Wauters
published in: Journal of Algebra
publication date: 1999-07-20T00:00:00Z
review text: Let \(K[G]\) denote the group algebra of the polycylic-by-finite group \(G\) over the field \(K\), and assume that \(K[G]\) is prime. Let \(S\) be a multiplicatively closed set of nonzero central elements of \(K[G]\) and, if \(K\) is absolute, assume that \(S\) contains an element not in \(K\). This paper obtains necessary and sufficient conditions for the localization \(K[G]_S\) to be a primitive ring. Specifically, the criterion is: (1) \(A=K\langle S,S^{-1}\rangle\) is a \(G\)-domain, (2) \((Q(ZK[G]):Q(A))\) is finite, and (3) \(J(K[G]_S)=0\), where the semiprimitivity condition (3) is needed only when \(G\) is Abelian-by-finite. It follows that \(K[G]_{ZK[G]}\) is a primitive ring provided that \(ZK[G]>K\) when \(K\) is absolute.
Mathematics Subject Classification ID: 16U20
Mathematics Subject Classification ID: 20F19
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 1295424
zbMATH Keywords: polycyclic-by-finite groups
zbMATH Keywords: prime group algebras
zbMATH Keywords: localizations
zbMATH Keywords: primitive rings
reviewed by: D. S. Passman
MaRDI profile type: Publication
full work available at URL: jabr.1998.7720
OpenAlex ID: W1984283189
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DOI: 10.1006/JABR.1998.7720
description: scientific article; zbMATH DE number 1295424