Torsion Modules Annihilators

Question

Let $R$ be a PID, let $B$ be a torsion $R$-module, and let $p$ be a prime in $R$. Show that if $pb = 0$ for some nonzero $b \in B$, then $\operatorname{Ann}(B)$ is a subset of $(p)$.

Well we can let $\operatorname{Ann}(B) = (t)$ for some $t \in R$ since it is an ideal in $R$. How to make use of the fact that $B$ is torsion (i.e. there is r in R such that rm=0 for each m in M) to prove the desired result?

Answer 1

Hint: Since $\operatorname{Ann}(B)\subseteq\operatorname{Ann}(b)$, it suffices to show $\operatorname{Ann}(b)=(p)$.

A stronger hint is hidden below.

Since $pb=0$, $(p)\subseteq\operatorname{Ann}(b)$. That is, $\operatorname{Ann}(b)$ is an ideal which contains $(p)$. What ideals are there which contain $(p)$?