If Ann(n)=P where P is a prime ideal, then there exists $a \in R-P$ such that $an=0$

Question

Suppose $R$ is a commutative ring and $N$ is a $R$-module. If it is the case the $P=Ann(n)$ for some $n\in N$ where $P$ is a prime ideal. Then there exists $a \in R-P $ such that $an=0$.

Hi all,

I am trying to prove this statement, but I have having a few problems. Partly because I am not sure how to start and I think the statement is false.

Since, $R/P \cong Rn$ as R-modules. Wouldn't it be the case that $an=0$ if only if $a \in P$?

Answer 1

You're right that something is wrong.

There can't exist an $a\notin Ann_R(n)=\{r\in R\mid rn=0\}$ such that $an=0$.