# If $R$ is a domain and $M$ a finitely generated $R$-module, is it true that $\bigcap_{f\in M^{*}}\ker{f}=\operatorname{Tor}M$?

Let $R$ be a domain and $M$ a finitely generated $R$-module. Let $M^{*}=\hom_{R}(M,R)$. Let Tor$M$ be the torsion submodule of $M$. It it true that $$\displaystyle\bigcap_{f\in M^{*}}\ker{f}=\operatorname{Tor}M\ ?$$

$(\supset)$ is an easy inclusion. I can't seem to prove $(\subset)$ for the general finitely generated case. I have done vector spaces and free modules. I can not think of any counter examples or find any in books or online. Does anyone know if this is true? Any hints toward it's proof if true would be appreciated. Thanks.

• Use the fact that $M^* \cong (M/\text{Tor}(M))^*$ to reduce to the case where $M$ is torsionfree, and show that the intersection is $0$, using the linked question to produce an injection of $M$ into a finite free module – zcn Aug 19 '14 at 18:52