A Generalization of Posner's Theorem on Derivations in Rings

In this paper, we generalize the Posner's theorem on derivations in rings as follows: Let R be an arbitrary ring, P be a prime ideal of R, and d be a derivation of R. If [[x, d(x)], y] is an element of P for all x, y is an element of R, then d(R) subset of P or R/P is commutative. In particular, if R is semiprime and d is a centralizing derivation of R, we prove that either R is commutative or there exists a minimal prime ideal P of R such that d(R) subset of P. As a consequence, we show that for any semiprime ring with a centralizing derivation there exists at least a minimal prime ideal P such that d(P) subset of P.