I'm doing some independent study with a professor in Ring/Field theory (I'm an undergraduate) and I have been having a hell of a time wrapping my head around problems involving Noetherian rings and modules. Here's an example of a problem I can't figure out:
Let $R$ be a left Noetherian ring, $_{R}M$ an $R$-module, and $m\in M$. Show $Rm$ is a left Noetherian $R$-Module.
I went about showing that $Rm$ was a submodule of $_RM$ and I feel decent about it. I'm really having trouble showing that $R$ left Noetherian tells us anything about whether $Rm$ is as well.
In general, is it standard practice in any problem involving Noetherian rings (modules) to apply Zorn's Lemma to say that it has maximal ideals (submodules)? Any intuition or help in this arena would be greatly appreciated.
HINT:
The left $R$-module $R$ is noetherian. The left $R$-module $Rm$ is an image of the noetherian left $R$-module $R$. Now apply a result about images of noetherian modules.