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Centralizers on prime and semiprime rings. (English) Zbl 0889.16016
Let $$R$$ be an associative ring. An additive mapping $$T\colon R\to R$$ is a left centralizer if $$T(xy)=T(x)y$$ holds for all $$x,y\in R$$. The main result asserts that if $$R$$ is a $$2$$-torsion free ($$2x=0\Rightarrow x=0$$, $$x\in R$$) non-commutative semiprime ring and $$S,T$$ are left centralizers such that $$[[S(x),T(x)],S(x)]=0$$ for each $$x\in R$$ ($$[a,b]=ab-ba$$), then $$[S(x),T(x)]=0$$ for each $$x\in R$$. If, moreover, $$R$$ is a prime ring and $$S\neq 0$$, then there exists $$\lambda\in C$$ (the extended centroid of $$R$$) such that $$T=\lambda S$$.

##### MSC:
 16U70 Center, normalizer (invariant elements) (associative rings and algebras) 16N60 Prime and semiprime associative rings 16W25 Derivations, actions of Lie algebras 16W10 Rings with involution; Lie, Jordan and other nonassociative structures
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