On $\tau$-centralizers of semiprime rings

Let $R$ be a semiprime 2-torsion free ring, and let $\tau$ be an endomorphism of $R$. Under some conditions we prove that a left Jordan $\tau$-centralizer of $R$ is a left $\tau$-centralizer of $R$. Under the same conditions we also prove that a Jordan $\tau$-centralizer of $R$ is a $\tau$-centralizer of $R$. We thus generalize Zalar's results to the case of $\tau$-centralizers of $R$.