Commuting and centralizing mappings in prime rings

Abstract:

Let $R$ be a ring. A mapping $F:R \to R$ is said to be commuting on $R$ if $[F(x),x] = 0$ holds for all $x \in R$. The main purpose of this paper is to prove the following result, which generalizes a classical result of E. Posner: Let $R$ be a prime ring of characteristic not two. Suppose there exists a nonzero derivation $D:R \to R$, such that the mapping $x \mapsto [D(x),x]$ is commuting on $R$. In this case $R$ is commutative.