Commutativity preserving maps revisited.
type: Item
instance of: scholarly article
title: Commutativity preserving maps revisited.
zbMATH Open document ID: 1142.16020
author: Matej Brešar
published in: Israel Journal of Mathematics
publication date: 2008-05-27T00:00:00Z
review text: The author refines some results of \textit{K. I. Beidar} and \textit{Y.-F. Lin} [Proc. R. Soc. Edinb., Sect. A, Math. 134, No. 6, 1023-1040 (2004; Zbl 1074.16019)] on the structure of commutativity preserving maps. Given any additive map of rings \(\alpha\colon A\to B\), set \(F(x)=[\alpha(x^2),\alpha(x)]\). The main result considers an additive \(\alpha\colon J\to Q\) where \(J\) is a Jordan ring and \(Q\) is an associative unital algebra with center \(C\), a field with \(\text{char\,}C\neq 2\). The author proves that if \(F(x)=0\) on \(J\), and if \(\alpha(J)\) is 3-free in \(Q\), then either \(\alpha=\mu\beta+\lambda\) for \(\mu\in C\), \(\beta\colon J\to Q\) a Jordan homomorphism, and \(\lambda\colon J\to C\) additive, or else a certain defined ideal of \(J\) is sent into \(C\) by \(\alpha\). By analyzing this ideal of \(J\) when \(J\) is associative, some corollaries of the main result are obtained. One such is that for \(A\) and \(B\) prime rings, neither of \(\text{char\,}2\) and neither satisfying the standard identity \(S_4\), then any additive and surjective \(\alpha\colon A\to B\) with \(F(A)=0\) must be \(\alpha=\mu\beta+\lambda\), as above, or \(\alpha(I)=0\) for a nonzero ideal \(I\) of \(A\). When \(A\) and \(B\) are centrally closed over \(C\) with \(\text{char\,}C\neq 2\), \(A\) does not satisfy \(S_4\), and \(\alpha\) is bijective and linear with \(F(A)=0\), then again \(\alpha=\mu\beta+\lambda\). The first of these corollaries has a version for \(A\) and \(B\) prime with involution and \(\alpha\) surjective with \(F(S(A))=0\) for \(S(A)\) the set of symmetric elements in \(A\).
reviewed by: Charles Lanski
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16U80
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 5279146
zbMATH Keywords: commutativity preserving maps
zbMATH Keywords: additive maps
zbMATH Keywords: prime algebras
zbMATH Keywords: symmetric elements
zbMATH Keywords: Jordan rings
zbMATH Keywords: Jordan homomorphisms
zbMATH Keywords: prime rings with involution
MaRDI profile type: Publication
full work available at URL: s11856-007-0100-0
OpenAlex ID: W2049114406
cites work: Rings with involution
cites work: On functional identities and commuting additive mappings
cites work: FUNCTIONAL IDENTITIES REVISED: THE FRACTIONAL AND THE STRONG DEGREE
cites work: Applying functional identities to some linear preserver problems.
cites work: On Functional Identities in Prime Rings with Involution II*
cites work: On functional identities andd-free subsets of rings, I
cites work: On functional identities andd-free subsets of rings. II
cites work: On surjective linear maps preserving commutativity
cites work: Q4865491
cites work: Commuting Traces of Biadditive Mappings, Commutativity-Preserving Mappings and Lie Mappings
cites work: Ond-Free Rings
cites work: Commutativity preserving linear maps on central simple algebras.
cites work: Q5639810
cites work: Q4112071
cites work: The Specht property of the ideals of identities of certain simple nonassociative algebras
cites work: Jordan homomorphisms of associative algebras with involution
cites work: On the relationship of a ring and the subring generated by its symmetric elements
cites work: Rings with involution and polynomial identities
cites work: Preserving commutativity
cites work: Q3910639
cites work: Linear maps that preserve commuting pairs of matrices
cites work: Q3947798
DOI: 10.1007/S11856-007-0100-0
description: scientific article; zbMATH DE number 5279146