Relative torsionfreeness and Frobenius extensions

Let $S\mathrm{/}R$ be a Frobenius extension with ${}_R{S}_R$ centrally projective over $R$. We show that if ${}_R\omega$ is a Wakamatsu tilting module then so is ${}_{S}S{\otimes }_R\omega$, and the natural ring homomorphism from the endomorphism ring of ${}_R\omega$ to the endomorphism ring of ${}_{S}S{\otimes }_R\omega$ 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.