Prime ideals and generalized derivations with central values on rings

Abstract

Our goal in the present paper is to study a connection between the commutativity of rings and the behaviour of its generalized derivations. More specifically, we investigate commutativity of quotient rings R/P where R is any ring and P is a prime ideal of R which admits generalized derivations satisfying certain algebraic identities acting on prime ideal P without the primeness (semi-primeness) assumption on the considered ring. This approach allows us to generalize some well known results characterizing commutativity of rings.