Principal ideals containing an ideal in a Noetherian integral domain

Question

Let $R$ be a Noetherian integral domain and $I$ a nonzero ideal consisting only of zero divisors on $R/(x)$, where $x$ is a nonzero element of $I$. Could we always find an element $y\notin (x)$ such that $yI\subseteq (x)$?

Thanks for any help!

Answer 1

Isn't this obvious?

$I$ is contained in the union of associated primes of $R/(x)$, so there is such a prime $\mathfrak p$ with $I\subset\mathfrak p$. Now write $\mathfrak p=\operatorname{Ann}(\hat y)$ for some non-zero $\hat y\in R/(x)$, and you are done.