Regularity conditions on skew and symmetric elements in superalgebras with superinvolution
type: Item
instance of: scholarly article
title: Regularity conditions on skew and symmetric elements in superalgebras with superinvolution
zbMATH Open document ID: 1196.17005
author: Jesús Laliena
author: Sara Sacristán
published in: Journal of Algebra
publication date: 2010-06-02T00:00:00Z
review text: One of the methodologies in the study of associative algebras with involution is to get results on the algebra from properties on the set of symmetric elements (or from the set of skew-symmetric elements). In some cases the algebraic properties of the set \(H\) of symmetric elements or the set \(K\) of skew-symmetric elements, determine the algebraic structure of the algebra. Many researchers have adhered to this philosophy along the history of modern algebra. J. M. Osborn, for instance, proved in 1967 that if \(A\) is a simple algebra of characteristic not 2 in which every nonzero symmetric element is invertible, then \(A\) must either be a division algebra or the algebra of \(2\times 2\) matrices over a field. Later, he generalized this to the case of a semisimple algebra \(A\) which is \(2\)-torsion free. If every symmetric element in \(A\) is either nilpotent or invertible, then \(A\) is either a division algebra or a direct sum of a division algebra and its opposite, or a \(2 \times 2\) matrices algebra over a field. In the work under review the authors are interested in studying Osborn's theorems (and related works concerning the skew-symmetric elements) in associative superalgebras with superinvolution. More precisely they prove that if \(A\) is a semiprime superalgebra with superinvolution \(*\) such that every nonzero homogeneous symmetric element is invertible then either \(A\) is a division superalgebra or \(A=B\oplus B^*\) where \(B\) is a division superalgebra and the superinvolution in \(A\) is the exchange superinvolution. They also prove the following result on nonsemiprime superalgebras: let \(A\) be a superalgebra with superinvolution such that every nonzero homogeneous symmetric element is invertible. If \(A\) has a nonzero nilpotent ideal then there exists a maximal nilpotent ideal \(N\) such that \(N^6=0\) and \(A/N\) is either a field of the direct sum of two copies of a field. Some other interesting results include skew versions of Osborn's th…
Mathematics Subject Classification ID: 17A70
Mathematics Subject Classification ID: 16W55
zbMATH DE Number: 5716390
zbMATH Keywords: associative superalgebra with superinvolution
zbMATH Keywords: Osborn's theorems
reviewed by: Candido Martín González
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.02.022
OpenAlex ID: W2039713928
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DOI: 10.1016/J.JALGEBRA.2010.02.022
description: scientific article; zbMATH DE number 5716390