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An identity related to centralizers in semiprime rings. (English) Zbl 1014.16021
Let $$R$$ be an associative ring. Recall that an additive mapping $$T\colon R\to R$$ is called a left (right) centralizer if $$T(xy)=T(x)y$$ ($$T(xy)=xT(y)$$) for all $$x,y\in R$$. The author proves that if $$R$$ is a $$2$$-torsionfree semiprime ring and $$T\colon R\to R$$ is an additive mapping such that $$2T(x^2)=T(x)x+xT(x)$$ for each $$x\in R$$, then $$T$$ is a left and right centralizer.

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