Structure of a quotient ring R/P and its relation with generalized derivations of R

The fundamental aim of this paper is to investigate the structure of a quotient ring R/P where R is an arbitrary ring and P is a prime ideal of R. More precisely, we will characterize the commutativity of R/P via the behavior of generalized derivations of R satisfying algebraic identities involving the prime ideal P. Moreover, various wellknown results characterizing the commutativity of prime (semi-prime) rings have been extended. Furthermore, examples are given to prove that the restrictions imposed on the hypothesis of the various theorems were not superfluous.