Relative torsionfreeness and Frobenius extensions

Let $S/R$ be a Frobenius extension with $_RS_R$ centrally projective over $R$. We show that if $_R\omega$ is a Wakamatsu tilting module then so is $_SS\otimes_R\omega$, and the natural ring homomorphism from the endomorphism ring of $_R\omega$ to the endomorphism ring of $_SS\otimes_R\omega$ is a Frobenius extension in addition that pd$(\omega_T)$ is finite, where $T$ is the endomorphism ring of $_R\omega$. We also obtain that the relative $n$-torsionfreeness of modules is preserved under Frobenius extensions. Furthermore, we give an application, which shows that the generalized G-dimension with respect to a Wakamatsu module is invariant under Frobenius extensions.