$A$-modules - An equality $I(M/N)=M/N$

Question

I am dealing with the following assertion:

Let $M$ be an $A$-module, $N$ a submodule of $M$ and $I$ an ideal of $A$ s.t. $M=IM+N$. We have $I(M/N)=M/N$.

I am in circuling thoughts. Given $m+N\in M/N$, so $m+N\in M=IM+N$...

It seems easy, but I cannot conclude.

Many thanks in advance!!

Answer 1

You have $I(M/N)=(IM+N)/N$. Your hypothesis is that $M=IM+N$. Therefore $$I(M/N)=(IM+N)/N=M/N$$

Note: this hypothesis is in connection with Nakayama's lemma.