Let $A$ be a commutative ring and $M$ an $A$-module. Let $I$ be any ideal of $A$. We have an epimorphism $M \otimes_A I \rightarrow IM$. It seems to me that this is not in general an isomorphism.
Q1: Any counterexample?
If $M$ is flat, then $M \otimes_A I \cong IM$. However, flatness seems to be a too strong condition for this equality to be true for any ideal $I$. I am interested in finding a characterization of $M$ such that $M \otimes_A I \cong IM$ for any ideal $I$ of $A$.
Q2: Any suggestions?