Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question

1 Answer 1

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.

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.