Let $R$ be a ring, $I$ an ideal of $R$, $M$ a finitely-generated $R$-module, and $N$ a submodule of $M$ such that $M = N + IM$.
The text I am reading asserts that it is trivial to show: $(N+IM)/N \cong (IM)/(IM \cap N)$. Trivial or not, I'm struggling to do it. How do we prove this?