Annihilator of Generalized Derivations with Power Values in Rings and Algebras

Abstract

Let F, G be two generalized derivations of prime ring R with characteristic different from 2 with associated derivations D1 and D2 respectively. We use the symbols C = Z(U) and U to denote the the extended centroid of R and Utumi ring of quotient of R respectively. Let 0 6= a ∈ R and F and G satisfy a{(F(xy) + G(yx))m − [x, y] n} = 0 for all x, y ∈ J , a nonzero ideal, where m and n are natural numbers. Then either R is commutative or there exists c, b ∈ U such that F(x) = cx and G(x) = bx for all x ∈ R.