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I have managed to prove that the aforementioned subrings are in bijection with the subsets of the primes, however, I haven't been able to prove that they are all noetherian. I need help.

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    $\begingroup$ Try to prove that they are all PIDs. $\endgroup$ – Regenbogen Mar 27 '10 at 0:59
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    $\begingroup$ This sounds a lot like a homework question, and is likely to be closed. $\endgroup$ – Theo Johnson-Freyd Mar 27 '10 at 1:45
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I will break this up into two steps, each of which is a standard exercise:

1) Let $R$ be a principal ideal domain with fraction field $K$. Every overring of $R$ -- i.e., every ring $S$ with $R \subset S \subset K$ -- is the localization of $R$ at a multiplicative subset.

2) If $R$ is a Noetherian ring and $S$ is a multiplicative subset, then the localization $S^{-1} R$ is a Noetherian ring.

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