Group-theoretic and topological invariants of completely integrally closed Prüfer domains

Abstract

We consider the lattice-ordered groups Inv$(R)$ and Div$(R)$ of invertible and divisorial fractional ideals of a completely integrally closed Prüfer domain. We prove that Div$(R)$ is the completion of the group Inv$(R)$, and we show there is a faithfully flat extension $S$ of $R$ such that $S$ is a completely integrally closed Bézout domain with Div$(R)\cong$ Inv$(S)$. Among the class of completely integrally closed Prüfer domains, we focus on the one-dimensional Prüfer domains. This class includes Dedekind domains, the latter being the one-dimensional Prüfer domains whose maximal ideals are finitely generated. However, numerous interesting examples show that the class of one-dimensional Prüfer domains includes domains that differ quite significantly from Dedekind domains by a number of measures, both group-theoretic (involving Inv$(R)$ and Div$(R)$) and topological (involving the maximal spectrum of $R$). We examine these invariants in connection with factorization properties of the ideals of one-dimensional Prüfer domains, putting special emphasis on the class of almost Dedekind domains, those domains for which every localization at a maximal ideal is a rank one discrete valuation domain, as well as the class of SP-domains, those domains for which every proper ideal is a product of radical ideals. For this last class of domains, we show that if in addition the ring has nonzero Jacobson radical, then the lattice-ordered groups Inv$(R)$ and Div$(R)$ are determined entirely by the topology of the maximal spectrum of $R$, and that the Cantor-Bendixson derivatives of the maximal spectrum reflect the distribution of sharp and dull maximal ideals.

arXiv commentMinor revision, including change of title. To appear in J. Pure Appl. Algebra