Dividend in Quotient Module Is Noetherian If Both The Quotient And Divisor Are

Proof

Let $M$ be an $R$-module with submodule $N$. Let $L$ be an arbitrary submodule of $M$. We will prove that $L$ is finitely generated, and thus $M$ is Noetherian.

By the third isomorphism theorem we have that

$(L+N)/N$ is a submodule of $M/N$ (the one corresponding with $L+N\le M$ under the correspondence theorem) and thus finitely generated. As such, $L/(L\cap N)$ must be finitely generated, and hence suppose it is generated by $\{y_1+L\cap N,\dots ,y_n+L\cap N\}$. However $L\cap N$ is also finitely generated since it is a submodule of the Noetherian module $N$. As such, suppose $L\cap N$ is generated by $\{x_1,\dots ,x_k\}$.

We then have that $L$ is generated by $\{y_1,\dots ,y_n,x_1,\dots ,x_k\}$.