On derivations in prime rings and Banach algebras

Abstract:

Let $R$ be a ring with center $Z(R)$. A mapping $F:R \to R$ is said to be centralizing on $R$ if $[F(x),x] \in Z(R)$ holds for all $x \in R$. The main purpose of this paper is to prove the following result, which generalizes a classical result of Posner: Let $R$ be a prime ring of characteristic not 2, 3, and 5. Suppose there exists a nonzero derivation $D:R \to R$ , such that the mapping $x \rightarrowtail [[D(x),x],x]$ is centralizing on $R$ . In this case $R$ is commutative. Combining this result with some well-known deep results of Sinclair and Johnson, we generalize Yood’s noncommutative extension of the Singer-Wermer theorem.