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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.

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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