Invertible and divisorial ideals of generalized Dedekind domains

Abstract

In this paper we give an ideal-theoretical characterization of a distinguished class of Prüfer domains, the class of generalized Dedekind domains. Namely, we prove that a Prüfer domain R is generalized Dedekind if and only if the divisorial ideals of R are exactly the ideals of type JP l … P n , where J is an invertible fractional ideal and P l , …, P n are (incomparable) nonzero prime ideals of R. We also show that, when R is a generalized Dedekind domain, the group of fractional invertible ideals of R is isomorphic to the free abelian group generated by the set of nonzero prime ideals of R and a basis for it is given by a suitable set of two-generated ideals with prime radical.