ON COMMUTATIVITY OF SEMIPRIME RINGS WITH GENERALIZED DERIVATIONS

Abstract

Let R be an associative ring. An additive mapping f : R -> R is called a generalized derivation if there exists a derivation d : R -> R such that f (xy) f (x)y + xd(y), for all x, y E R. In this paper, we explore the commutativity of semiprime rings admitting generalized derivations f and g such that one of the following holds for all x, y is an element of R. Let (f, d) and (g, h) be two generalized derivations of R. (i) f (x)y = xg(y), (ii) f [(x, y])=-/+[x, y], (iii)f(xoy)=-/+ xoy for all x, y is an element of R.