In an algebra lecture we looked at the following lemma:
Let $R$ be a ring, $M$ an $R$-module and $N \subset M$ a submodule. Then
$N$ and $M/N$ are finitely generated $\implies$ $M$ is also finitely generated.
I understand the proof for this lemma and it makes sense to me.
Our professor also mentioned the converse:
$M$ finitely generated $\implies$ $M/N$ finitely generated
$M$ finitely generated + $R$ noetherian $\implies$ $N$ and $M/N$ finitely generated
Here I don't understand why we don't have:
$M$ finitely generated $\implies$ $N$ and $M/N$ finitely generated
Since $N$ is a submodule of $M$ and therefore $(N,+)$ is a subgroup of $(M,+)$, I would have intuitively said that $N$ also must be finitely generated. But this must be false since we need the condition that $R$ is noetherian for that to be the case.
I would greatly appreciate any help.