Let $M$ be a module over the commutative ring $R$ and suppose that $N$ and $H$ are submodules of $M$ such that $M/N$ and $M/H$ are both Noetherian.
Show that $M/(N \cap H)$ is Noetherian.
I took any arbitrary submodules $K/(N \cap H)$, I must show this is finitely generated. From here $K$ is a submodule of $M$. From here how can I say $K/(N\cap H)$ is finitely generated?
Best Answer
There is an injective module homomorphism from $M/(N\cap H)$ to $(M/N)\oplus(M/H)$ so $M/(N\cap H)$ is isomorphic to a submodule of $(M/N)\oplus(M/H)$. Each submodule of a Noetherian module is Noetherian, and $(M/N)\oplus(M/H)$ is Noetherian.