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