Some identities in quotient rings

Let R be an associative ring, P a prime ideal of R: In this paper, we study the structure of the ring R=P and describe the possible forms of the generalized derivations satisfying certain algebraic identities on R: As a consequence of our theorems, we first investigate strong commutativity preserving generalized derivations of prime rings, and then examine the generalized derivations acting as (anti)homomorphisms in prime rings. Some commutativity theorems also given in semi-prime rings.