This is a homework problem.
Suppose, M is a finitely generated torsion module over a PID R. Then, M is uniserial if and only if M has only one elementary divisor.
So far I was able to prove the backward implication, as for $R/(p^e)$ all possible submodules are given by $(p^{e-i})/(p^e)$. For the forward implication, I figured that there can't be two non-associate primes. Since the submodules $R/(p^e)$ and $R/(q^f)$ have distinct annihilators (considered as isomorphic copies inside M), so none of them can contain the other. I am stuck with the part why there can't be powers of same prime. I thank you in advance for your help.