Do we have some good examples of local subrings of number fields which are not valuation rings?


Do we have an easy criterion for determining whether a local subring of a number field is a valuation ring? There are certainly many such criteria on Wikipedia and Atiyah-Macdonald. What I want to ask for is a criterion which use in an essential way the condition of being a subring of a number ring.



When is a local subring of a number field a valuation ring?
A localization of a Dedekind ring at a (non-zero) prime is a Dedekind ring with a unique non-zero prime, hence finitely-many, hence is a principal ideal domain with a unique non-zero prime...


