2023/04/07 by Lorenzo Guerrieri, Guerrieri, Lorenzo, K. Alan Loper +1
Mathematics · Medicine · #13A15 #13A18 #13B02 #13B21 #13B30 #13F05 #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Magnolia and Illicium research #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2304.03723
openalex publication_date 2023/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is well-known that an integrally closed domain D can be express as the intersection of its valuation overrings but, if D is not a Prüfer domain, the most of valuation overrings of D cannot be seen as localizations of D. The Kronecker function ring of D is a classical construction of a Prüfer domain which is an overring of D[t], and its localizations at prime ideals are of the form V(t) where V runs through the valuation overrings of D. This fact can be generalized to arbitrary integral domains by expressing them as intersections of overrings which admit a unique minimal overring. In this article we first continue the study of rings admitting a unique minimal overring extending known results obtained in the 70's and constructing examples where the integral closure is very far from being a valuation domain. Then we extend the definition of Kronecker function ring to the non-integrally closed setting by studying intersections of Nagata rings of the form A(t) for A an integral domain admitting a unique minimal overring.