Do not understand a corollary of $N$,$N/M$ are Noetherian submodules of $M$ iff $M$ is noetherian,

Question

Do not understand a corollary of: "$N$,$N/M$ are Noetherian submodules of $M$ iff $M$ is noetherian"

My professor said as a corollary of the above statement that: "If $R$ is a left Noetherian every finitely generated $R$-module is left Noetherian."

But I can not see why this is a corollary of the above statement, could anyone explain this for me please?

Also could anyone give me a hint for the proof of this corollary?

Thanks!

Answer 1

Let $R$ be Noetherian. Then $R^n$ is Noetherian; this is proved by induction. $R^n$ has a submodule $N$ with $N\cong R$ and $R^n/N\cong R^{n-1}$ etc.

If $M$ is finitely generated, then it is isomorphic to a quotient of some $R^n$, which is Noetherian.