TITLE: Are submodules of finitely generated free modules always free?
QUESTION [14 upvotes]: I don't believe it's true that all submodules of a finitely generated free module are free just based on results from google. Is there a canonical simple example of a finitely generated free module with a submodule that is not free?

REPLY [5 votes]: A very simple example is $\mathbb{Z}_6$ as a $\mathbb{Z}_6$-module is free (generated by $[1]$) and its ideal (regarded as a submodule) $2 \mathbb{Z}_6=\{[0],[2],[4]\}$ has no basis since $[2][3]=[4][3]=[0]$