Supercentralizing automorphisms on prime superalgebras.
type: Item
instance of: scholarly article
title: Supercentralizing automorphisms on prime superalgebras.
zbMATH Open document ID: 1188.16039
author: Yu Wang
published in: Taiwanese Journal of Mathematics
publication date: 2009-11-30T00:00:00Z
review text: The author proves a superalgebra version of [\textit{J. H. Mayne}, Can. Math. Bull. 27, 122-126 (1984; Zbl 0537.16029)]. Here \(A\) is a superalgebra over a commutative ring \(K\) with \(1/2\in K\). The super-commutator, \([a,b]_s\), of \(a,b\in A\) is the extension of \([a,b]_s=ab-ba\) when \(a\) or \(b\) is an even element, and \([a,b]_s=ab+ba\) when both are odd. Let \(Z_s(A)=\{a\in A\mid [a,b]_s=0\) for all \(b\in A\}\), and \(\varphi\in\Aut(A)\) that preserves the grading. The first main result assumes that \(A\) is noncommutative and prime (the product of nonzero graded ideals is not zero) and that \([\varphi(a),a]_s\in Z_s(A)\) for all \(a\in A\) and proves that \(\varphi=I_A\) (the identity map), and if \(A\) has elements of odd degree then \(A\) is an order in a quaternion algebra. Assuming instead that \([\varphi(a),a]_s=0\) for all \(a\in A\) forces \(A\) to have no odd degree elements, and then if \(A\) is not commutative, again \(\varphi=I_A\).
reviewed by: Charles Lanski
Mathematics Subject Classification ID: 16W55
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 5640595
zbMATH Keywords: prime superalgebras
zbMATH Keywords: automorphisms
zbMATH Keywords: supercentralizing maps
zbMATH Keywords: supercommuting maps
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.11650/twjm/1500405551
OpenAlex ID: W4251654408
DOI: 10.11650/TWJM/1500405551
description: scientific article; zbMATH DE number 5640595