I've been trying to think of an $R$-module that is Noetherian, not finite and is not a ring.
Examples that I know are:
1 A finite Abelian group is a Noetherian $\mathbb Z$-module (of course it satisfies a.c.c. because it's finite as a set)
2 $\mathbb Z$ is a Noetherian $\mathbb Z$-module. It's submodules correspond to ideals $I$ and $\mathbb Z $ is a PID. So every chain of ideals will eventually end in prime ideals. (the chain might branch into several maximal ideals)
3 Similar to 2, $k[x]$ where $k$ is a field is also a PID and by the same argument as in 2 also Noetherian.
But two of these three are rings and one is finite.
What are more interesting examples of Noetherian $R$-modules?
And is every PID (=principal ideal domain) a Noetherian module?