Primes and annihilators

Let $R$ be a PID and $M$ be a torsion $R$ -module. Suppose $p m = 0$ for some nonzero $m \in M$ and prime element $p \in R$ . Prove that $Ann (M) \subseteq ⟨ p ⟩$ .

Hints

The ideal $Ann (M) \subseteq R$ is principal. Also, since $p \in R$ is prime and $R$ is a PID, for every element $a \in R$ we either have that $p$ divides $a$ (i.e., $a \in ⟨ p ⟩$ ), or $a$ and $p$ are relatively prime (in which case we can write $r a + s p = 1_{R}$ for some $r, s \in R$ ).