Do we have $M\otimes_A(I_1/I_2)\cong I_1M/I_2M$?

Question

Let $A$ be a commutative ring with unity, $I_1,I_2$ two ideals of $A$ with $I_2\subset I_1$, $M$ an $A$-module. Do we have $M\otimes_A(I_1/I_2)\cong I_1M/I_2M$?

Answer 1

No, the statement fails in general. Consider $A = \mathbb Z$, $M = \mathbb Z/(2)$ and $I = (2)$. Then $IM$ is trivial whereas $I \otimes_A M$ is not. (Here $I_2$ is taken to be $0$.)

Note that $I \otimes_A M$ is isomorphic to $IM$ if $M$ is flat.