I'm reading a proof of Nakayama's theorem; it says at a certain step that:
For $M$, a finitely generated module on a ring $R, N$ a submodule, and $I$ an ideal of the ring $R$:
If $M = N + IM$, then $M/N = I(M/N)$.
I am definitely sure that this question might seem stupid, but I still can't convince myself about that implication.
The proof hints at the fact that for $x$ say representing a certain class in $M/N$ there is $x'$ in $M$ such as $x+N = I (x'+N)$, but again, isn't that equivalent to saying that $M=IM$?
Is there an isomorphism between $IM/N$ and $I(M/N)$?
I know that I am confused so I thank you in advance for clarifying this for me.