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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.

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  • $\begingroup$ And your attempts? $\endgroup$ – Don Thousand Mar 13 at 16:39
  • $\begingroup$ @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. $\endgroup$ – David Hubbard Mar 13 at 16:43
  • $\begingroup$ What is $ a $$?$ $\endgroup$ – Bernard Mar 13 at 19:02
  • $\begingroup$ @Bernard a is a general non zero element in P $\endgroup$ – David Hubbard Mar 13 at 19:22
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Hint:

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

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