Let $A$ be a commutative ring, and $M$ be an $A$-module, and $M^*$ be $\mathrm{Hom}_A(M,A)$. Let $f$ be the map from $M \otimes_A M^*$ to $\mathrm{Hom}_A(M,M)$, such that, for all $x=\sum_i a_i \otimes b_i \in M \otimes_A M^*$, $f(x)$ is the homomorphism $y \in M \mapsto \sum_ib_i(y)a_i \in M$. Is it true that $f$ is always a monomorphism ? If not, is there a necessary and sufficient condition on $M$ for $f$ to be a monomorphism ?