Seeking for an example of a non-Noetherian module having a Noetherian submodule.

Question

We know that a $R$-module $M$ is Noetherian if a submodule $N$ and its Quotient module $M/N$ is Noetherian.

Now I want a counter example where a submodule is Noetherian but the original module is not. Please any help will be appreciated.

Answer 1

If the zero module of any not-Noetherian module was not a satisfying example, then you can easily get a nonzero example by considering the unique minimal subgroup of a Prüfer $p$-group for a prime $p$.

The elements of order $p$ form the unique minimal submodule considering the group as a $\mathbb Z$ module, and the module is not Noetherian because there is a tower of submodules, each submodule consisting of elements with order no greater than $p^n$ for some $n$.