# A natural homomorphism of dual modules which is not a monomorphism?

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.

-
Related discussion math.stackexchange.com/questions/69184/… –  Ehsan M. Kermani Mar 11 '12 at 0:39
Dear Buble, Did you try any examples? E.g. with $A = \mathbb Z$ and $M = N$? Regards, –  Matt E Mar 11 '12 at 4:28