A note on generalized derivations on prime rings

Abstract

Let R be a prime ring with the extended centroid C and symmetric Martindale quotient ring Q_s(R). In this paper we prove the following result. Let F: R rightarrow R be a generalized derivation associated with a non-zero derivation d on R and let h be an additive map of R such that F(x)x=xh(x) for all xin R. Then either R is commutative or F(x)=xp and h(x)=px where pin Q_{s}(R).