A Result on Prime Rings with Generalized Derivations

Abstract

Abstract In this paper we investigate the following result. Let R be a prime ring, Q its symmetric Martindale quotient ring, C its extended centroid, I a nonzero ideal of R. If F and G are the two generalized derivation of R such that (F(xy) + G(yx))n − (xy ∓ yx)n = 0, for all x, y ∈ I, then either R is commutative or F (x) = x, G(x) = ∓x for all x ∈ R and n = 1.