If $M$ is a free module finitely generated, are all it's submodule finitely generated?

Question

Let $R$ a ring. I firstly thought that all submodule of a finitely generated module where finitely generated, but as I asked here, it's wrong. But is it true if a module is free ? In particular if $\displaystyle M\cong\bigoplus_{i=1}^n R$ and if $N\subsetneq M$, then $\displaystyle N\cong \bigoplus_{i=1}^m R$ for an $m<n$ ?

Answer 1

No, this is not true. Take any non-Noetherian ring $R$. Then $R$ is a free $R$-module of rank $1$, but it has an ideal (i.e. submodule) which is not finitely generated.

For a specific example, choose your favorite field $k$ and take $R=k[x_1,x_2,\dots]$, and the ideal $I=(x_1,x_2,\dots)$.