study of the structure of quotient ringssatisfying algebraic identities

Assuming that r is an associative ring with prime ideal p, this paper investigates the commutativity of the quotient ring r/p, as well as the possible forms of generalized derivations satisfying certain algebraic identities on r. we give results on strong commutativity, preserving generalized derivations of prime rings, using our theorems. finally, an example is given to show that the restrictions on the ideal p are not superfluous.