In an integrally closed, Noetherian, local, integral domain of dimension $1$, the maximal ideal $P$ is eventually principal

Question

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.

Answer 1

Hint:

The quotient ring $R_P/aR_P$ is a local noetherian ring of dimension $0$ so $PR_P/aR_P$ is its nilradical.