Let $R$ be a DVR. This means that it is a PID with a unique maximal ideal. Let us denote $m$ to be the maximal ideal. Then there is a unique irreducible element $\pi$ up to multiplication my a unit such that $m=(\pi)$.
I know that in a PID all maximal ideals are either $(0)$ or $(p)$ where $p\in R$ is prime. Does this denn immediately mean that $\pi$ needs to be zero or prime?
I think this should be true since every PID is a UFD and in a UFD every irreducible element is prime so $\pi$ should be prime or am I wrong?
Thanks for your help.
Remark PID=principal ideal domain, UFD=unique factorization domain, DVR=discrete valuation ring