Shahrood University of Technology Journal of Algebraic Systems 2345-5128 12 2 2023 01 01 AN IDENTITY RELATED TO θ-CENTRALIZERS IN SEMIPRIME RINGS 367 377 2963 10.22044/jas.2023.11856.1607 EN Abbas Zivari-Kazempour Department of Mathematics, Ayatollah Borujerdi University, Borujerd, Iran. Journal Article 2022 04 24 ‎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 $T‎: ‎R\longrightarrow R$‎ <br />‎is an additive mapping such that‎<br />‎$‎‎$‎<br />‎T(xyx)=θ(x)T(y)θ(x)‎,<br />‎$‎‎$‎<br />‎holds for all $x‎, ‎y\in R$‎, ‎then‎ <br />‎$T$ is a $θ$-centralizer‎<br />either $R$ is unital‎ or $θ(Z(R))=Z(R)$. semiprime ring‎ ‎centralizer‎ ‎$\theta$-centralizer‎ ‎epimorphism https://jas.shahroodut.ac.ir/article_2963_c3bceba354de13e81079166621748469.pdf