Isomorphisms of integral alternative loop rings
type: Item
instance of: scholarly article
title: Isomorphisms of integral alternative loop rings
zbMATH Open document ID: 0696.17015
DOI: 10.1007/BF02844273
author: Edgar G. Goodaire
author: C. Polcino Milies
publication date: 1988-01-01T00:00:00Z
review text: Let us call a - not necessarily associative - ring A left or right alternative iff \(x^ 2y=x(xy)\) or \((xy)y=xy^ 2\) for all \(x,y\in A\), resp., and alternative iff it is left and right alternative. - For an associative ring R with unity and a loop L the loop ring of L over (the coefficient ring) R is constructed precisely in the same way as a group ring and denoted by RL. The investigations under review are concerned with sufficient conditions on L such that the right alternativity of RL implies the alternativity of RL. By handling some nontrivial term transformations the authors succeed in proving the following results: for finite L and char \(R\neq 2\), if RL is right alternative it is alternative; if L is a Moufang loop the following statements are equivalent: RL is left alternative, RL is right alternative, RL is alternative. - The authors conjecture more generally that right alternativity of a loop ring always implies its alternativity.
Mathematics Subject Classification ID: 17D05
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 17D15
zbMATH DE Number: 4140001
zbMATH Keywords: right alternative algebras
zbMATH Keywords: alternative algebras
zbMATH Keywords: Moufang loop
zbMATH Keywords: loop ring
reviewed by: Wilfried Lex
MaRDI profile type: Publication
cites work: Q3248988
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published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 4140001