Abstract:The purpose of this paper is to investigate identities satisfied by centralizers on prime and semiprime rings. We prove the following result: Let $R$ be a noncommutative prime ring of characteristic different from two and let $S$ and $T$ be left centralizers on $R$. Suppose that $[S(x),T(x)]S(x)+S(x)[S(x),T(x)]=0$ is fulfilled for all $x\in R$. If $S\not =0$ $(T\not =0)$ then there exists $\lambda $ from the extended centroid of $R$ such that $T=\lambda S$ $(S=\lambda T)$.
Keywords: prime ring, semiprime ring, extended centroid, derivation, Jordan derivation, left (right) centralizer, Jordan left (right) centralizer, commuting mapping, centralizing mapping
AMS Subject Classification: 16A12, 16A68, 16A72