$\tau: S^{-1} (\operatorname{Hom}_A(M, N)) → \operatorname{Hom}_{S^{-1}A}(S^{-1}M, S^{-1}N)$ is injective if $M$ is a finitely generated $A$-module 
Let $A$ be a ring.  Let $M$ and $N$ be two $A$-modules, $S$ is a multiplicatively closed subset of $A$. Show that there is a homomorphism of $S^{-1}A$-modules $$\tau: S^{-1} (\operatorname{Hom}_A(M, N)) → \operatorname{Hom}_{S^{-1}A}(S^{-1}M, S^{-1}N)$$ and this is injective if $M$ is a finitely generated $A$-module. Is it true if $M$ is not finitely generated?

To give an "counter example" to the claim in case $M$ is not finitely generated I guess one can take $A=\mathbb{Z}, S=\mathbb{Z}\backslash \{0\}, N = M = \bigoplus_{n\geq 2}\mathbb{Z}/n\mathbb{Z}$, but I am not sure how to show that the map is not injective. For proving the main claim I have no clue. Can anyone give me some hints? Thank you in advance.

Hint. In your example $S^{-1}M=0$, but $S^{-1}\operatorname{Hom}_A(M,N)\ne0$.
