# Quotient Noetherian module

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?

## 1 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.