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.
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