## On centralizers of semiprime rings.(English)Zbl 0746.16011

The author proves two results on mappings of a semi-prime 2-torsion free ring $$R$$. An additive map $$T\colon R\to R$$ is a left (right) centralizer if $$T(xy)=T(x)y$$ ($$T(xy)=xT(y)$$) for all $$x,y\in R$$, and is a Jordan centralizer if $$T(xy+yx)=T(x)y+yT(x)=xT(y)+T(y)x$$ for all $$x,y\in R$$. The first main result proves that if $$T(x^2)=T(x)x$$ for all $$x\in R$$, then $$T$$ is a left centralizer. The second shows that any Jordan centralizer of $$R$$ is both a left and right centralizer.

### MSC:

 16N60 Prime and semiprime associative rings 16W10 Rings with involution; Lie, Jordan and other nonassociative structures 16W20 Automorphisms and endomorphisms 16U70 Center, normalizer (invariant elements) (associative rings and algebras)
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