TITLE: Is every submodule of cyclic module over PID cyclic?
QUESTION [10 upvotes]: I needed to check if any subgroup of cyclic abelian group is cyclic and successfully proved that "yes" using the fact that $\mathbb{Z}$ is Euclidian domain. It's easy to give an example that it doesn't hold for modules over arbitrary ring, but I am pretty sure that it does for PID. How can I prove it?

REPLY [4 votes]: This was answered in the comments, but I will write it out here to mark this as "answered" :)
Every cyclic $R$-module is of the form $R/I$ for some ideal $I$ of $R$. Submodules correspond to ideals and vice versa. Every ideal of $R/I$ is of the form $J/I$ for some ideal $J$ in $R$, and since $J$ is principal (generated by $x$, say), $J/I$ is also principal (generated by $x + I$), i.e., it is cyclic.