Let $R$ a commutative local principal ideal ring with 1 that is not Artinian. So it's Krull dimension is non zero. Let $P\subsetneq M$ a prime ideal and M the maximal ideal of R, since P and M are principal ideals let $P=(p)$ and $M=(m)$ for some $p,m\in R$.
It is true that exists $s\in \mathbb{N}$ such that $p=m^s$?
My approach:
Since $p\in M$ there's a $r_1\in R$ such that $p=r_1m$. We have two options $r_1\in M$ o didn't.
If $r_1\notin M$ we have that $m=r_1^{-1}p$ a contradiction.
If $r_1\in M$ we can find a that $p=r_2m^2$ and so on...
I don't know if this process ends, because the non zero krull dimension could be infinte, there's some hypothesis missing or something that I'm not considering?