2015/12/10 by Olivier A. Heubo-Kwegna, Heubo-Kwegna, Olivier A., Bruce Olberding +3
Mathematics · #13A05 #13A15 #13F05 #Commutative Algebra (math.AC) #FOS: Mathematics #Rings, Modules, and Algebras #math.AC #msc:13A05 #msc:13A15 #msc:13F05
paper · pdf · doi:10.48550/arxiv.1512.03312
Minor revision, including change of title. To appear in J. Pure Appl. Algebra
openalex publication_date 2015/12/10 · arxiv created 2016/05/03 · arxiv updated 2016/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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) ≅ 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.