TITLE: Example of a finitely generated module with submodules that are not finitely generated
QUESTION [35 upvotes]: 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!

REPLY [5 votes]: Regarding the "hint" above. Let $f \in I$. Then $f$ has no constant term and has only finitely many of the variables $X_i$. Now suppose that $f_1,\dots,f_k \in I$ generate $I$ over $A$, in other words, suppose that $I$ is finitely generated over $A$. Let $n$ be a large enough natural number so that none of the $f_i$ contains the variable $X_n$. Now $X_n \in I$ therefore $X_n= p_1f_1 + \ldots + p_kf_k$. In this expression, set $X_1= \ldots = X_{n-1}=0$ and $X_n=X_{n+1}=\ldots = 1$. One gets that $0=1$, a contradiction.