Let R be a ring with identity. An R-module M is Artinian if it satisfies the descending chain condition on submodules. What is an example of an Artinian module with a proper submodule that is not finitely generated?
There you can find a concrete example of an Artinian module over the integers which is non-Noetherian and hence has a non-finitely generated submodule.
In the example at hand, the module itself is not finitely generated, and there are also lots of non-finitely generated proper submodules.