I was given this question, where $A$ is a commutative ring:
If every prime ideal of $A$ is maximal, then the chain ${}\dots \leq x^2 A \leq x A$ stabilizes, for all $x \in A$.
Also, the clue:
Consider $S_x = \{ x^n(1-xy) : n \in \mathbb{N} ,\, y \in A \}$. If $\, 0\notin S_x$, then get a maximal ideal $P$ with the property $P \cap S_x = \varnothing$.
I was considering the fact that, if $x$ is nilpotent, then the chain stabilizes trivially. Therefore, we can assume otherwise, and get a prime ideal $P$ such that $x \notin P$, and therefore it is maximal by hypothesis.
However, I am unable to derive anything else from this. I have thought about taking the quotient of the chain by $P$, but I haven't been able to conclude.