Minimal primes and zero divisors

Question

Let $R$ be a commutative local ring, $M$ a finitely generated $R$-module, and $x \in M$. Is it true that if for any $p \in$ $\operatorname{Min}(R)$ there exists $a_{p}\notin{p}$ such that $a_{p}x=0$, then $x=0$ ?

Answer 1

No. Let $Ass(R)\neq Min(R)$ and $R=M$. note that $⋃_{p∈Ass R}p=zd(R)$ (the set of zerodivisors of R)
example:
$R=k[X,Y]/(XY,Y^2)$. then $Min(R)=(\bar Y)$ , $\bar X$ $\notin(\bar Y)$ and $\bar X$.$\bar Y = 0$ but $\bar Y \neq 0$
(here $\bar Y$ is your question's $x$ and $\bar X$ is your question's $a_{p}$ and $(\bar Y)$ is your question's $p$)