Let $R$ be an integrally closed, Noetherian, local, integral domain of dimension 1 with unique maximal ideal $P$.
Take an element $a \in P$ that is non zero. Show that for some $n$, $P^n$ is contained in $aR$ (the ideal generated by $a$).
The point of this exercise was to prove unique factorisation in a Dedekind Domain. A step in this is to show that the localisation of $R$ with respect to $P$ is a PID.