Is every submodule of cyclic module over PID cyclic?
Solution 1:
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.