Let R be a PID, B a torsion R module and p a prime in R. Prove that if $pb=0$ for some non zero b in B, then $\text{Ann}(B)$ is a subset of (p)

Question

I have tried to show it but to no avail. Its an exercise in chapter 12.1 of Dummit and Foote which I am learning by myself. Thanks in advance for any help.

Answer 1

The assumption says that the ideal $\operatorname{Ann}(b):=\{r\in R\mid rb=0\}$ contains $(p)$. Since in a PID, every prime generates a maximal ideal and $\operatorname{Ann}(b)$ is clearly not the whole ring ($1\not\in \operatorname{Ann}(b)$) we must thus have equality $(p)=\operatorname{Ann}(b)$. Clearly $\operatorname{Ann}(B)$ is contained in $\operatorname{Ann}(b)$.