AN IDENTITY RELATED TO θ-CENTRALIZERS IN SEMIPRIME RINGS

‎Let $R$ be a $ 2$-torsion-free semiprime ring and $\theta$ be an epimorphism of $R$‎. ‎In this paper‎, ‎under special hypotheses‎, we prove that if $T‎: ‎R\longrightarrow R$‎ ‎is an additive mapping such that‎‎$‎‎$‎‎T(xyx)=θ(x)T(y)θ(x)‎,‎$‎‎$‎‎holds for all $x‎, ‎y\in R$‎, ‎then‎ ‎$T$ is a $θ$-centralizer‎either $R$ is unital‎ or $θ(Z(R))=Z(R)$.