Certain Pairs of Generalized Derivations of Semiprime Rings

Abstract

Let R be a semiprime ring with the cancellation property ,U be a set of R, (D,d) and (G,g) be generalized derivations of R ,if R admits to satisfied some conditions ,where d acts as a left centralizer (resp.g acts as a left centralizer).Then there exist idempotents ϵ1, ϵ 2, ϵ 3 C and an invertible element λ C such that ϵ iϵ j= 0, for i≠j, ϵ 1+ ϵ 2+ ϵ 3 =1,and ϵ 1f(x)=λϵ 1g(x), ϵ 2g(x)= 0, ϵ 3f(x)= 0 hold for all x U. (*)During our paper we will using (*)for denoted to above result.