I've been mixing up my reading with a little category theory. When thinking about dual modules, this idea popped up.
Suppose $M$ and $N$ are modules over some commutative ring $A$. Suppose $M^\vee$ denotes the dual module of $M$, that is, $M^\vee=\operatorname{Hom}_A(M,A)$.
Is there an example of such $M$ and $N$ so that the natural homomorphism $M^\vee\otimes N\to\text{Hom}_A(M,N)$ is not a monomorphism (not injective)? Thanks.