On a generalization of the notion of centralizing mappings

Abstract:

Let $R$ be a ring with center $Z$. A mapping $f:R \to R$ is called centralizing (resp. commuting) if $[f(x),x] \in Z$ (resp. $[f(x),x] = 0$) for all $x \in R$. In this paper we consider a more general case where a mapping $f:R \to R$ satisfies $[[f(x),x],x] = 0$ for all $x \in R$; it is shown that if $R$ is a prime ring of characteristic not 2, then every additive mapping with this property is commuting.