On additive isomorphisms of prime rings preserving polynomials
type: Item
instance of: scholarly article
title: On additive isomorphisms of prime rings preserving polynomials
zbMATH Open document ID: 0934.16030
published in: Journal of Algebra
publication date: 2000-04-09T00:00:00Z
review text: A Lie (Jordan, respectively) isomorphism between (associative) rings is an additive isomorphism that preserves a polynomial \(xy-yx\) (\(xy+yx\), respectively). The paper considers additive isomorphisms between prime rings that preserve any multilinear polynomial in noncommuting variables. This of course covers and unifies both the classical cases of Lie and Jordan isomorphisms, as well as some other special cases treated in the literature. A simplified version of the main result is: Let \(B\) and \(A\) be prime rings, \(C\) be the extended centroid of \(A\) and \(F\) be a subring of \(C\) with 1 such that \(B\) is an \(F\)-algebra. Further, let \(m\geq 2\) and \(f(x_1,\ldots,x_m)\in F\langle X\rangle\), the free \(F\)-algebra on the set \(X=\{x_1,x_2,\ldots\}\), be a nonzero multilinear polynomial. Suppose that \(\gamma\colon B\to A\) is an isomorphism of \(F\)-modules such that \(\gamma(f(b_1,\ldots,b_m))=f(\gamma(b_1),\ldots,\gamma(b_m))\) for all \(b_1,\ldots,b_m\in B\). If \(\text{char}(A)\neq 2\) and \(A\) contains elements that are not algebraic of degree \(\leq 2m-1\) over the extended centroid, then \(\gamma\) is of the form \(\gamma(b)=t\beta(b)+\nu(b)\), where \(t\in C\), \(t\neq 0\), \(\beta\) is an (anti-)monomorphism of \(B\) into the central closure of \(A\) and \(\nu\) is an additive map of \(B\) into \(C\). The proof rests heavily on some recent progress made in the theory of functional identities.
reviewed by: Matej Brešar
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 1340565
zbMATH Keywords: Lie isomorphisms
zbMATH Keywords: additive isomorphisms
zbMATH Keywords: prime rings
zbMATH Keywords: multilinear polynomials
zbMATH Keywords: Jordan isomorphisms
zbMATH Keywords: extended centroids
zbMATH Keywords: functional identities
author: K. I. Beidar
author: Y. Fong
MaRDI profile type: Publication
full work available at URL: jabr.1998.7833
OpenAlex ID: W1970701588
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DOI: 10.1006/JABR.1998.7833
description: scientific article; zbMATH DE number 1340565