Let $\mathscr{N}(R)$ denote the set of all *nilpotent* elements in a ring $R$.

I have actually done an exercise which states that if $x \in \mathscr{N}(R)$, then $x$ is contained in every prime ideal of $R$.

The converse of this statement is: if $x \notin \mathscr{N}(R)$, then there is a prime ideal which does not contain $x$.

But I am not able to prove it. Can anyone provide me a proof of this result?