Canonical homomorphism related to ideal is an isomorphism

Question

I have a problem to do the exercise 1.2.1 b on Liu. Namely,

Let $M$ be an $A$-module, $I\subseteq \operatorname{Ann}(M)$ an ideal, $N\ne M$ is an $A$-module such that $I\subseteq \operatorname{Ann}(N)$. Why does the canonical homomorphism $M\otimes_AN\to M\otimes_{A/I}N$ is an isomorphism?

Answer 1

This comes from $M\otimes_{A}N\simeq M\otimes_{A/I}(A/I\otimes_{A}N)$, and $A/I\otimes_{A}N\simeq N/IN=N$.