Reversible group rings.
type: Item
instance of: scholarly article
title: Reversible group rings.
zbMATH Open document ID: 1068.16033
published in: Journal of Algebra
publication date: 2004-11-22T00:00:00Z
review text: An associative ring \(R\) with identity is said to be reversible if \(ab=0\) implies \(ba=0\) for all \(a,b\in R\). The main problem of this paper is the question when the group ring \(K[G]\) of a group \(G\) over a field \(K\) is reversible. The authors prove that if \(G\) is a non-Abelian torsion group and \(K[G]\) is reversible, then \(G\) is Hamiltonian. Thus, using the structure theorems for Hamiltonian groups and for group rings of finite Abelian groups over fields, they show that the question can be reduced to group rings \(K[Q_8]\), where \(Q_8\) is the quaternion group, and \(p=\text{char}(K)\) must be \(0\) or \(2\). It is proved that if \(p\neq 2\) (resp., \(p=2\)), then \(K[Q_8]\) is reversible if and only if the equation \(x^2+y^2+1=0\) (resp., \(x^2+x+1=0\)) has no solution in \(K\). Let \(G\) be a non-Abelian torsion group and let \(K\) be a field of characteristic \(p\). The main result asserts that \(K[G]\) is reversible if and only if \(G\) is Hamiltonian and \(p=0\) (resp., \(p=2\) and \(G/Q_8\) has no element of even order) and the equation \(x^2+y^2+1=0\) (resp., \(x^2+x+1=0\)) has no solution in any cyclotomic field \(K(\xi_n)\) for any odd \(n\) which is the order of an element of \(G\).
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 2117268
zbMATH Keywords: reversible rings
zbMATH Keywords: group rings
zbMATH Keywords: Hamiltonian groups
zbMATH Keywords: zero divisor conjecture
author: Marin Guţan
author: Andrzej Kisielewicz
reviewed by: S. V. Mihovski
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.02.011
OpenAlex ID: W1982721680
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DOI: 10.1016/J.JALGEBRA.2004.02.011
description: scientific article; zbMATH DE number 2117268