## Joso Vukman

*Centralizers on prime and semiprime rings *

Comment.Math.Univ.Carolinae 38,2 (1997) 231-240. **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

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