Assume $R$ is a ring, $\psi:M\rightarrow N$ a surjective $R$-modules homomorphism. Then, $M$ is noetherian $\iff$ $N$ and $K=\operatorname{Ker}(\psi)$ are noetherian.
So I proved $\leftarrow$. Now I assume $M$ is noetherian. It's not hard to show $N$ is noetherian since $\psi$ is surjective. Now I tried to prove that $K$ is noetherian - I took $K'\subseteq K$ a submodule of $K$. So, $K'\subseteq M$ and therefore it's a submodule in $M$ and so if finitely generated. Is my reasoning correct or am I missing something (maybe the fact that it's finitely generated in $M$ doesn't mean its finitely generated in $K$?).
Any help would be appreciated.