In the paper "P-orderings and polynomial functions on arbitrary subsets of Dedekind rings" Bhargava defines a Dedekind ring to be any Noetherian, locally principal ring in which all nonzero primes are maximal.
Which is a locally principal ring? It is a ring $R$ for which $R/I$ is a principal ideal ring for any ideal $I$?
Locally principal ring means each localisation at a prime ideal is a principal ideal ring. In the definition I know for a Dedekind ring, it is an integral domain. In such a case, this means each localisation at a maximal ideal is a discrete valuation domain.