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Keywords:
prime ring; semiprime ring; extended centroid; derivation; Jordan derivation; left (right) centralizer; Jordan left (right) centralizer; commuting mapping; centralizing mapping
prime ring; semiprime ring; extended centroid; derivation; Jordan derivation; left (right) centralizer; Jordan left (right) centralizer; commuting mapping; centralizing mapping
Summary:
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 $S\neq 0$ $(T\neq 0)$ then there exists $\lambda $ from the extended centroid of $R$ such that $T=\lambda S$ $(S=\lambda T)$.
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