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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.

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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