Invertible and nilpotent elements in the group algebra of a unique product group.
type: Item
instance of: scholarly article
title: Invertible and nilpotent elements in the group algebra of a unique product group.
zbMATH Open document ID: 1189.16023
author: Erhard Neher
published in: Acta Applicandae Mathematicae
publication date: 2009-12-09T00:00:00Z
review text: Let \(k\) be a commutative ring with unity, \(G\) a group and \(kG\) its group algebra. Kaplansky's problem asks whether the group ring of a torsion-free group over a zero-divisor free ring is also zero-divisor free. There is a well-known result which states that the crossed product of a u.p. group (which is torsion-free) over a zero-divisor free ring is also zero-divisor free and has trivial units only. The author applies tools of algebraic geometry to describe the upper nil radical (group of units) of \(kG\) in case \(KG\) is zero-divisor free (has trivial units only) for every field \(K\) which is a \(k\)-algebra, conditions satisfied for a u.p. group. It is proved that the upper nil radical is the group algebra of \(G\) over the nil radical of \(k\), which coincides with the set of all nilpotent elements, and units decompose into the sum of trivial units according to a direct sum decomposition of the ring \(k\) plus a nilpotent element.
reviewed by: János Kurdics
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16N40
zbMATH DE Number: 5644660
zbMATH Keywords: unique product groups
zbMATH Keywords: u.p. groups
zbMATH Keywords: group rings
zbMATH Keywords: trivial units
zbMATH Keywords: nilpotent elements
zbMATH Keywords: nil radical
zbMATH Keywords: zero-divisor free rings
MaRDI profile type: Publication
full work available at URL: s10440-008-9370-8
OpenAlex ID: W2077990538
cites work: Q3843687
cites work: Q3999650
cites work: UNITS AND ISOMORPHISM IN GROUP RINGS
cites work: Q4145843
cites work: Q4198893
DOI: 10.1007/S10440-008-9370-8
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description: scientific article; zbMATH DE number 5644660