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 $t‎: ‎rlongrightarrow r$‎ ‎is an additive mapping such that‎‎$‎‎$‎‎t(xyx)=θ(x)t(y)θ(x)‎,‎$‎‎$‎‎holds for all $x‎, ‎yin r$‎, ‎then‎ ‎$t$ is a $θ$-centralizer‎either $r$ is unital‎ or $θ(z(r))=z(r)$.