# Finitely generated module with a submodule that's not finitely generated

Can someone give an example of a ring R, a left R-moudule M and a submodule N of M s.t M is finitely generated, but N is not finitely generated ?

I tried a couple of examples of modules I know and got nothing...

-
 – Jack Schmidt Mar 27 '12 at 14:18

Consider the simplest possible nontrivial (left) $R$-module: $R$ itself. It's certainly finitely generated, by $\{ 1 \}$. The submodules are exactly the (left) ideals of $R$. So you want a ring which has (left) ideals which are not finitely generated. For example, you could use a non-Noetherian commutative ring, such as $\mathbb{Z}[X_1, X_2, X_3, \ldots ]$.
Even more is true: A ring $R$ is noetherian if and only if submodules of f.g. modules over $R$ are f.g. – Martin Brandenburg Mar 27 '12 at 10:46