On $\theta$-centralizers of semiprime rings (II)

The following result is proved: Let $R$ be a 2-torsion free semiprime ring, and let $T\colon R\to R$ be an additive mapping, related to a surjective homomorphism $\theta\colon R\to R$, such that $2T(x^2)=T(x)\theta(x)+\theta(x)T(x)$ for all $x\in R$. Then $T$ is both a left and a right $\theta$-centralizer.