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Commutativity preserving mappings on semiprime rings. (English) Zbl 0865.16015

Let \(R\) and \(R'\) be rings. By a commutativity preserving map we mean a map \(\theta:R\to R'\) with the property that \([\theta(x),\theta(y)]=0\) whenever \([x,y]=0\). A simple example is a map of the form \(\theta(x)=c\phi(x)+\zeta(x)\) where \(c\in R'\) is a central element, \(\phi\) is an isomorphism or an antiisomorphism, and \(\zeta\) is a map into the center of \(R'\). One usually wants to prove that on certain rings this example is essentially the only possible example of a bijective additive map preserving commutativity. A number of results of this kind have been obtained over the last twenty years, primarily using linear algebra and operator theory techniques. There is a similar problem which has mostly been considered in ring theory, namely, the problem of describing the form of Lie isomorphisms (which, of course, also preserve commutativity) and Lie derivations.
It has turned out that for rather large classes of rings all these (and some similar) problems can be solved using a unified approach based on a characterization of commuting traces of biadditive maps (that is, maps satisfying \([B(x,x),x]=0\) for all \(x\in R\), where \(B:R\times R\to R\) is a biadditive map). For prime rings this was done by the reviewer [in Trans. Am. Math. Soc. 335, No. 2, 525-546 (1993; Zbl 0791.16028)]. In the paper under review, this approach is used to obtain analogous results for semiprime rings. (Also submitted to MR).

MSC:

16N60 Prime and semiprime associative rings
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16W20 Automorphisms and endomorphisms
16W25 Derivations, actions of Lie algebras
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras

Citations:

Zbl 0791.16028
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References:

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