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

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.

• And your attempts? – Don Thousand Mar 13 at 16:39
• @Don Thousand not sure if this counts as an attempt but presumably the noetherian condition should be of use. I imagine assuming that the powers of P are never contained in aR allow you to construct an increasing chain of ideals which must terminate. – David Hubbard Mar 13 at 16:43
• What is $a$$?$ – Bernard Mar 13 at 19:02
• @Bernard a is a general non zero element in P – David Hubbard Mar 13 at 19:22

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