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Centralizers on semiprime rings. (English) Zbl 1057.16029
Let \(R\) be an associative ring. An additive mapping \(T\colon R\to R\) is called a centralizer if it is both a left and a right centralizer, i.e. if \(T(xy)=T(x)y\) and \(T(xy)=xT(y)\) for all \(x,y\in R\). If \(R\) is an arbitrary ring and \(T\) is a centralizer, then \(T(xyx)=xT(y)x\) for all \(x,y\in R\). The main result of the paper states that the converse holds whenever \(R\) is a 2-torsion free semiprime ring.

MSC:
16W20 Automorphisms and endomorphisms
16N60 Prime and semiprime associative rings
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16R50 Other kinds of identities (generalized polynomial, rational, involution)
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