I am going over some counterexamples for the the isomorphism $M^{*} \otimes M \rightarrow Hom_R( M,M)$. In particular I have been trying to understand what happens if you remove the various restrictions on the module conditions on $M$ necessary guarantee the bijectivity of the isomorphism. The strongest condition required that I know of $M$ be finitely generated projective so the natural question in search of a counterexample is formulated below:
Let $R$ be the ring $\mathbb{Z} / \mathbb{4Z}$ and let $I = 2 \mathbb{Z} / 4 \mathbb{Z}$ be an ideal of $R$. Consider the quotient $R/I$ as an $R$-module $M$.
Now let $\theta : Hom_R(M,R) \otimes M \rightarrow Hom_R(M,M)$ be the canonical mapping. That is the mapping given by $\theta(f\otimes m)(x)=f(x) m$ for all $f\in Hom_R(M,R)$, all $x \in M$ and all $m\in M$.
How do we show that $\theta$ is not one to one or onto?