Example of a finitely generated module with submodules that are not finitely generated

I'm looking for an example of a finitely generated module with submodules that are not finitely generated.

I've found a similar question dealing with group (i.e. an example of a finitely generated group with subgroups that are not finitely generated). But I can't figure out whether that question do help to this one.

And I actually want to find a more "module-like" example rather than an example derived from a 'strange' group.

Can you please help? Thank you!

The ideal $I=\langle X_1,X_2,...,X_n,... \rangle \subset \mathbb R[X_1,X_2,...,X_n,...]=A$ can be seen as a submodule of the free $A$-module of dimension one $A=A^1$, and that module is not finitely generated. Do you see why?
(Hint: even in a polynomial ring with infinitely many indeterminates, each polynomial involves only finitely many variables. In other words $\mathbb R[X_1,X_2,...,X_n,...]=\bigcup_{k\geq 1}\mathbb R[X_1,X_2,...,X_k] \;$ )

Here's a fairly simple example (of a non-Noetherian ring): the ring $R$ of polynomials in one indeterminate $X$ having rational coefficients but with an integer constant term. Its ideal of elements with zero constant term is not finitely generated as an $R$-module.