Some finiteness chain conditions on the set of intermediate rings
type: Item
instance of: scholarly article
title: Some finiteness chain conditions on the set of intermediate rings
zbMATH Open document ID: 1196.13007
author: Ahmed Ayache
published in: Journal of Algebra
publication date: 2010-06-08T00:00:00Z
review text: Let \(R \subset S\) be an extension of integral domains, \([R,S]\) be the set of rings between \(R\) and \(S\), and \(K\) be the quotient field of \(R\). Let \(\mathrm{Supp}(S/R) = \{P \in\) Spec\((R)| PS = S\}\) and \(\overline{R_S}\) be the integral closure of \(R\) in \(S\). We say that the extension \(R \subset S\) satisfies the finite chain condition (FCC) if each chain of distinct rings between \(R\) and \(S\) is finite. It is known that \(R \subset K\) satisfies FCC (i.e., \(R\) is an FC-domain) if and only if \(\overline{R}\), the integral closure of \(R\) in \(K\), is a Prüfer domain with finite spectrum, (ii) \(\overline{R}\) is a finite \(R\)-module, and (iii) \(R/(R:\overline{R})\) is an Artinian ring where \((R:\overline{R}) = \{x \in R| x\overline{R} \subseteq R\}\) [\textit{R. Gilmer}, Proc. Am. Math. Soc. 131, No. 8, 2337--2346 (2003; Zbl 1017.13009)]. The author of the paper under review noted that (i) if \(R = K\), then \(R \subset S\) satisfies FCC if and only if \(R \subset S\) is an extension of fields with finite degree and (ii) if \(R \neq K\) and \(R \subset S\) satisfies FCC, then \(R \subset S \subseteq K\). Let \(S \subseteq K\). The main result proved in this paper is that \(R \subset S\) satisfies FCC if and only if there is a finite maximal chain \(R = R_0 \subset R_1 \subset \cdots \subset R_n = S\) in \([R,S]\) such that one of the \(R_i\) is \(\overline{R_S}\), if and only if (i) \((\overline{R_S}, S)\) is a normal pair (i.e., each \(T \in [\overline{R_S}, S]\) is integrally closed in \(S\)) and \(\mathrm{Supp}(S/R)\) is finite, (ii) \(\overline{R_S}\) is a finite \(R\)-module, and (iii) \((R:\overline{R_S})\) is an intersection of finitely many maximal ideals of \(R\).
reviewed by: Gyu Whan Chang
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13B22
zbMATH DE Number: 5718200
zbMATH Keywords: finite chain condition
zbMATH Keywords: normal pair
zbMATH Keywords: finite maximal chain
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.03.009
OpenAlex ID: W2087078097
cites work: Residually algebraic pairs of rings
cites work: Intermediary rings in normal pairs
cites work: Overrings of Commutative Rings. III: Normal Pairs
cites work: Q4517574
cites work: Homomorphismes minimaux d'anneaux
cites work: Q5661989
cites work: Some finiteness conditions on the set of overrings of an integral domain
cites work: A lower bound for the number of intermediary rings
cites work: Q5603302
cites work: Q5735274
cites work: On minimal overrings of a noetherian domain
cites work: Pairs of Domains Where all Intermediate Domains are Noetherian
DOI: 10.1016/J.JALGEBRA.2010.03.009
Recommended article: On finiteness of chains of intermediate rings
Recommended article: Ring extensions with some finiteness conditions on the set of intermediate rings
Recommended article: A special chain theorem in the set of intermediate rings
Recommended article: Finiteness of the set of intermediary rings in normal pairs
Recommended article: Q3809950
Recommended article: Q3356455
Recommended article: Q3625111
Recommended article: On chains of intermediate rings resulting from the juxtaposition of minimal ring extensions
Recommended article: On some intermediate rings
Recommended article: Some results on rings with chain conditions
description: scientific article; zbMATH DE number 5718200