The torsion product property in alternative algebras
type: Item
instance of: scholarly article
title: The torsion product property in alternative algebras
zbMATH Open document ID: 0878.17030
author: Edgar G. Goodaire
author: C. Polcino Milies
published in: Journal of Algebra
publication date: 1998-01-05T00:00:00Z
review text: Let \(R\) be an alternative ring with unity. Then \({\mathcal U}(R)\) denotes the set of units of \(R\), and \(T{\mathcal U}(R)\) is the set of the units of finite order in \(R\). The ring \(R\) has the torsion product property (briefly, ``has t.p.p.'') if the product of torsion units is again a torsion unit, i.e. if \(T{\mathcal U}(R)\) is a subloop of \({\mathcal U}(R)\). The authors pose the question under which conditions an alternative ring has the torsion product property. For alternative division rings and vector matrix algebras the authors establish the following results: (1) Let \(D\) be a division ring of characteristic \(p>0\). Then for any integer \(n>1\) the full matrix ring \(M_n(D)\) has t.p.p. if and only if \(D\) is a field which is an algebraic extension of its prime subfield. (2) Let \(F\) be a field of characteristic \(p>0\). Then Zorn's vector matrix algebra \({\mathfrak Z}(F)\) has t.p.p. if and only if \(F\) is algebraic over its prime field \(\mathcal P\). The main topic of this paper is the study of the torsion product property in alternative loop rings which are closely related to RA loops. An RA loop \(L\) is a loop with the property that for any commutative and associative ring \(R\) with unity the loop ring \(RL\) is alternative, but not associative. Such a loop \(L\) is necessarily a Moufang loop and thus diassociative. For RA loops the following hold: Assume \(L\) is an RA loop with torsion subloop \(T\). (a) Let \(F\) be a field of characteristic 0. Then the loop ring \(FL\) has t.p.p. if and only if \(T\) is an abelian group, for each \(t \in T\) and each \(x \in L\) there is a positive integer \(i\) such that \(xtx^{-1}=t^i\), and for each non central element \(t\in T\), \(F\) contains no root of unity whose order is the order of \(t\). Moreover, the torsion units of \(FL\) are contained in the units of \(FT\). (b) Let \(F\) be a field of characteristic \(p > 0\) with prime field \(\mathcal P\). Then \(FL\) has t.p.p. if and only if…
Mathematics Subject Classification ID: 17D05
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 16U60
zbMATH DE Number: 934242
zbMATH Keywords: alternative ring
zbMATH Keywords: torsion units
zbMATH Keywords: alternative division rings
zbMATH Keywords: alternative loop rings
zbMATH Keywords: RA loops
zbMATH Keywords: Moufang loop
reviewed by: Huberta Lausch
MaRDI profile type: Publication
full work available at URL: jabr.1996.0249
OpenAlex ID: W1978531965
DOI: 10.1006/JABR.1996.0249
description: scientific article; zbMATH DE number 934242