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On certain equations satisfied by centralizers in rings. (English) Zbl 1158.16308

Summary: We investigate some equations with the left centralizers in prime and semiprime rings. We prove, for example, the following result: Let \(R\) be a 2- and 3-torsion free semiprime ring and let \(S,T\colon R\to R\) be left centralizers, such that \((S(x)T(x))^2=S(x)^2T(x)^2\) is fulfilled for all \(x\in R\). In this case \([S(x),T(x)]=0\) holds for all \(x\in R\). In case \(R\) is a noncommutative prime ring and \(S\neq 0\) (\(T\neq 0\)), then there exists \(\lambda\) from the extended centroid of \(R\), such that \(T=\lambda S\) (\(S=\lambda T\)).

MSC:

16W20 Automorphisms and endomorphisms
16N60 Prime and semiprime associative rings
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