On Jordan left-I-centralizers of prime and semiprime gamma rings with involution

Abstract

Let M be a 2-torsion free Γ-ring with involution I satisfying the condition xαyβz=xβyαz for all x,y,z∈M and α,β∈Γ. The object of our paper is to show that every Jordan left-I-centralizer on a semiprime Γ-ring with involution I, is a reverse left-I-centralizer. We solve some functional equations in prime and semiprime Γ-rings with involution I by means of the above results. Moreover, we discuss some more related results.