An Equations Related to θ-Centralizers on Lie Ideals

Abstract

The purpose of this paper is to prove the following result :
Let R be a 2-torsion free prime ring , U a square closed Lie ideal, and T,ï±: Rï‚®R are
additive mappings, such that 3T(xyx) = T(x)θ(yx) + θ(x)T(y)θ(x) + θ(xy)T(x) and
θ(x)T(xy+yx)θ(x) = θ(x)T(y)θ(x2) + θ(x2)T(y)θ(x) holds for all pairs x, y  U , if θ is a
surjective endomorphism on U, and T(u) U, for all uU, then T(xy)= T(x)θ(y) = θ(x)T(y)
for all x,yU.