Let $R$ be a ring, and $x\in R$ is a non-nilpotent element of $R$. Let $P$ be an ideal of $R$ that is maximal with respect to the property that $x^n \notin P $, $\forall n$. Show that $P$ is prime.
Note that $P$ may not be a maximal ideal, only maximal with respect to this property. I know that a non-nilpotent element must note be contained in some prime ideal, since the nilradical is the intersection of the prime ideals. However, I am not really sure how to approach this problem.