Integral Domains whose Simple Overrings are Intersections of Localizations

Topological properties of localizations, flat overrings and sublocalizations

We study the set of localizations of an integral domain from a topological point of view, showing that it is always a spectral space and characterizing when it is a proconstructible subspace of the space of all overrings. We then study the same problems in the case of quotient rings, flat overrings and sublocalizations.

非整闭的 Kronecker 函数环与具有唯一极小扩环的整环

Non-integrally closed Kronecker function rings and integral domains with a unique minimal overring

本文将 Kronecker 函数环的构造推广到非整闭整环，通过研究具有唯一极小扩环的整环及其 Nagata 环的交集，并构造整闭包远离赋值整环的新例子。

Well-centered overrings of an integral domain

Let A be an integral domain with field of fractions K. We investigate the structure of the overrings B of A (contained in K) that are well-centered on A in the sense that each principal ideal of B is generated by an element of A. We consider the relation of well-centeredness to the properties of flatness, localization and sublocalization for B over A. If B = A[b] is a simple extension of A, we prove that B is a localization of A if and only if B is flat and well-centered over A. If the integral closure of A is a Krull domain, in particular, if A is Noetherian, we prove that every finitely gene…

Gorenstein Prüfer 整环的上环刻画

An overring characterization of Gorenstein Prüfer domains

该研究刻画了 Gorenstein Prüfer 整环的等价条件，证明其每个上环为 Gorenstein 平坦模当且仅当该整环是 Gorenstein Prüfer 整环，且通过结合对偶性与导子论证完成证明并给出相关性质与实例区分。

Jaffard families and localizations of star operations

We generalize the concept of localization of a star operation to flat overrings; subsequently, we investigate the possibility of representing the set $\mathrm{Star}(R)$ of star operations on $R$ as the product of $\mathrm{Star}(T)$, as $T$ ranges in a family of overrings of $R$ with special properties. We then apply this method to study the set of star operations on a Prüfer domain $R$, in particular the set of stable star operations and the star-class groups of $R$.

Intersections of quotient rings and Pruefer v-multiplication domains

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 $\{S_λ\midλ\in Λ\}$ of overrings of $D$ is called a defining family of $D$, if $D = \bigcap\{S_λ\midλ\in Λ\}$. 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, …