2015/09/19 by El Baghdadi Said, Marco Fontana, Said, El Baghdadi +5
Mathematics · #13B30 #13E99} #13F05 #13G05 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics #Rings, Modules, and Algebras #math.AC #msc:13B30 #msc:13F05 #msc:13G05
paper · pdf · doi:10.48550/arxiv.1509.05884
arxiv created 2015/09/19 · openalex publication_date 2015/09/19 · arxiv updated 2015/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let D be an integral domain with quotient field K. Call an overring S of D a subring of K containing D as a subring. A family \Sλ|λ∈ Λ\ of overrings of D is called a defining family of D, if D = \bigcap\Sλ|λ∈ Λ\. Call an overring S a sublocalization of D, if S has a defining family consisting of rings of fractions of D. Sublocalizations and their intersections exhibit interesting examples of semistar or star operations. We show as a consequence of our work that domains that are locally finite intersections of Prüfer v-multiplication (respectively, Mori) sublocalizations turn out to be Prüfer v-multiplication domains (respectively, Mori); in particular, for the Mori domain case, we reobtain a special case of \cite[Théorème 1]Q and \cite[Proposition 3.2]de. We also show that, more than the finite character of the defining family, it is the finite character of the star operation induced by the defining family that causes the interesting results. As a particular case of this theory, we provide a purely algebraic approach for characterizing Prüfer v-multiplication domains as a subclass of the class of essential domains (see also \cite[Theorem 2.4]FT).