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.

1$\begingroup$ Try to prove that they are all PIDs. $\endgroup$– RegenbogenMar 27, 2010 at 0:59

5$\begingroup$ This sounds a lot like a homework question, and is likely to be closed. $\endgroup$– Theo JohnsonFreydMar 27, 2010 at 1:45
1 Answer
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.