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I am trying to prove the following implication, and can't seem to find my way around all the equivalent definitions of Dedekind domains and DVRs:

I have a ring $R$ with the following properties:

1) $R$ is Noetherian.

2) $R$ is integrally closed.

3) Every nonzero prime ideal in $R$ is maximal.

I wish to show that every localization of $R$ at a maximal ideal is a principal ideal domain.

Does anyone know a direct argument proving this (i.e. not passing through the myriad of equivalent definitions of Dedekind domains and DVRs)? Alternatively, I would be thankful if someone could provide me with a "road map" to proving this claim in a a way which would convince someone (namely, me) without knowledge of Dedekind domains and DVRs.

Thanks a lot!

Roy

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    I‘m not sure whether the proofs I have in mind would satisfy you. It seems to me that you quickly get to the statement “an integrally closed Noetherian domain with a unique non-zero prime ideal is in fact principal” and from there you have to do some real work, and you would be proving one direction of an equivalence between definitions. I think Serre does this in a very low-tech way at the beginning of his _Local Fields_, although I don't like that proof very much. Are willing to use some commutative algebra? Basic dimension theory really helps.2012-08-29
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    Related question: http://math.stackexchange.com/questions/183467/discrete-valuation-ring-associated-with-a-prime-ideal-of-a-dedekind-domainComments may only be edited for 5 minutes(click on this box to dismiss)2012-08-30

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