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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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