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I'm trying to solve this question from a book and I have already proved 1.

  1. Let $R$ be a local domain which is not a field. Suppose that the maximal ideal $M$ of $R$ is principal and satisfies $\cap_{n=1}^{\infty}M^n=0$. Show that every non-zero ideal of $R$ is a power of $M$, and hence that $R$ is a DVR.

  2. Deduce that a Noetherian valuation ring is either a field or a DVR.

kind thanks in advance

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Valuation rings have the property that for all elements $a,b$ we have $a|b$ or $b|a$. It follows easily that every finitely generated ideal is principal. In particular, a noetherian valuation ring has a principal maximal ideal. Besides, Krull's Theorem states that $\bigcap_{n \geq 0} \mathfrak{m}^n=0$ in a local noetherian ring $(R,\mathfrak{m})$. Therefore, 2 follows from 1.

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I guess this assumes the reader knows why the intersection equality in part 1 holds in a Noetherian valuation ring? – rschwieb Mar 14 '13 at 18:20
Ah yes, Krull's Theorem. – Martin Brandenburg Mar 14 '13 at 18:32

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