Let $R$ be a ring. We have:
Cohen's theorem for Rings. If all prime ideals of $R$ are finitely generated, then $R$ is Noetherian.
Now let $M$ be an $R$-module. We have
Cohen's Theorem for Modules. Let $M$ be finitely generated and further assume that $\mathfrak pM$ is finitely generated for each prime ideal $\mathfrak p$ in $R$. Then $M$ is Noetherian.
One can adopt the standard proof of Cohen's Theorem for Rings to prove Cohen's Theorem for Modules. But is there a short cut if we assume the former to prove the latter?