Annihilator of generalized derivations with power values in rings and Algebras

Let F , G be two generalized derivations of prime ring R with characteristic different from 2 with associated derivations D 1 and D 2 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 (cid:54) = 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