2008/09/17 by D. D. Anderson, David F. Anderson, Anderson, D. D. +5
Computer Science · Decision Sciences · Mathematics · #13A15 #13F05 #13G05 #Advanced Algebra and Logic #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Fuzzy and Soft Set Theory #Rings, Modules, and Algebras #math.AC #math.AG #msc:13A15 #msc:13F05 #msc:13G05
paper · pdf · doi:10.48550/arxiv.0809.2947
arxiv created 2008/09/17 · openalex publication_date 2008/09/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let ∗ be a star operation on an integral domain D. Let \f(D) be the set of all nonzero finitely generated fractional ideals of D. Call D a ∗--Prüfer (respectively, (∗, v)--Prüfer) domain if (FF-1)∗=D (respectively, (FvF-1)∗=D) for all F∈ \f(D). We establish that ∗--Prüfer domains (and (∗, v)--Prüfer domains) for various star operations ∗ span a major portion of the known generalizations of Prüfer domains inside the class of v--domains. We also use Theorem 6.6 of the Larsen and McCarthy book [Multiplicative Theory of Ideals, Academic Press, New York--London, 1971], which gives several equivalent conditions for an integral domain to be a Prüfer domain, as a model, and we show which statements of that theorem on Prüfer domains can be generalized in a natural way and proved for ∗--Prüfer domains, and which cannot be. We also show that in a ∗ --Prüfer domain, each pair of ∗ -invertible ∗ -ideals admits a GCD in the set of ∗ -invertible ∗ -ideals, obtaining a remarkable generalization of a property holding for the "classical" class of Prüfer v--multiplication domains. We also link D being ∗ --Prüfer (or (∗, v)--Prüfer) with the group Inv∗(D) of ∗ -invertible ∗ -ideals (under ∗-multiplication) being lattice-ordered.