I came across this solution that seeks to prove that any submodule of a cyclic module is cyclic.
Proof: Let $M$ be a cyclic module, so that $\phi:R \rightarrow M$ is a surjection under $\phi(r)=r \cdot m$ for some fixed $m$ . Therefore $M$ is isomorphic to $R/ \ker \phi$, i.e. a quotient of $R$ by some ideal. Further, any submodule of $M$ is isomorphic to an ideal of $R/ \ker \phi$. Since $R/ \ker \phi$ is a PID also, any submodule of it is generated by one element. Hence any submodule $N \subset M$ is cyclic.
My question is: How to prove that $R/\ker \phi$ is also a PID?
As pointed out in the comments, $R/I$ may not be a domain, however:
Let $I$ be any ideal of the principal ideal domain $R$. Let $\mathscr J$ be an ideal of $R/I$, let $\pi:R \to R/I$ be the canonical projection, viz. $\pi(r) = r + I$ for $r \in R$, and let $J = \pi^{-1} (\mathscr J)$. Then I claim that $J$ is an ideal of $R$ and $I \subset J$. The second assertion is pretty obvious: any $s \in I$ maps to $\bar 0 = 0 + I = I$ in $R/I$, and since $\bar 0 \in \mathscr J$, the set $\pi^{-1}(\bar 0) \subset J$ contains $s$; thus $I \subset J$. As for the first, it is easy to see that $J$ is an ideal of $R$, straight from the definitons: if $a, b \in J$, then $\pi(a - b) = \pi(a) - \pi(b) \in \mathscr J$ since $\pi(a), \pi(b) \in \mathscr J$; thus $a - b \in \pi^{-1}(\mathscr J) = J$; if $r \in R$, then $\pi(ra) = \pi(r) \pi(a) \in \mathscr J$, an ideal of $R/I$; this implies of course that $ra \in J$. Bearing these facts in mind we note that $J = (j)$ for some $j \in R$ since $R$ is a PID. Then any $l \in J = \pi^{-1}(\mathscr J)$ is of the form $pj$ for some $p \in R$. Thus $\pi(l) = \pi(pj) = \pi(p) \pi(j) = \bar p \bar j$; this shows that $\mathscr J = (\bar j) = (j + I)$; it follows that $R/I$ is a principal ideal ring, though not in general a domain. QED.