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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 \(T=\lambda S\).

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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