Vukman, Joso; Kosi-Ulbl, Irena On certain equations satisfied by centralizers in rings. (English) Zbl 1158.16308 Int. Math. J. 5, No. 5, 437-456 (2004). 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 \(T=\lambda S\) (\(S=\lambda T\)). Cited in 1 Document MSC: 16W20 Automorphisms and endomorphisms 16N60 Prime and semiprime associative rings Keywords:left centralizers; prime rings; semiprime rings; extended centroids PDFBibTeX XMLCite \textit{J. Vukman} and \textit{I. Kosi-Ulbl}, Int. Math. J. 5, No. 5, 437--456 (2004; Zbl 1158.16308)