Characterizing the ring extensions that satisfy FIP or FCP
type: Item
instance of: scholarly article
title: Characterizing the ring extensions that satisfy FIP or FCP
zbMATH Open document ID: 1271.13022
author: David E. Dobbs
author: Gabriel Picavet
author: Martine Picavet-L'Hermitte
published in: Journal of Algebra
publication date: 2013-05-27T00:00:00Z
review text: Let \(R\) and \(S\) be two commutative rings. The extension \(R \subseteq S \) is said to satisfy FIP (for the ``finitely many intermediate algebras property'') if the set of all \(R\)-subalgebras of \(S\) is finite. The extension \(R \subseteq S \) is said to satisfy FCP (for the ``finite chain property'') if each chain of \(R\)-subalgebras of \(S\) is finite. It is clear that each extension that satisfies FIP must also satisfy FCP; the converse is false, as can be easily seen via field-theoretic examples. In the present paper, the authors prove several (parallel) characterizations of the FIP and FCP properties. Note that a number of results about FCP are generalized from domains to arbitrary (commutative) rings. Among the main results of the paper, I would like to mention the following. (a) Let \(R \subseteq S \) be an extension of rings rings, with \(\overline R\) the integral closure of \(R \) in \(S\). Then \(R \subseteq S \) satisfies FIP (respectively, FCP) if and only if both \(R \subseteq \overline R \) and \(\overline R \subseteq S \) satisfy FIP (respectively, FCP). (b) If \(R\) is integrally closed in \(S\), then \(R \subseteq S \) satisfies FIP if and only if \(R \subseteq S \) satisfies FCP if and only if \((R, S)\) is a normal pair such that Supp\(_R (S/R)\) is finite. (c) If \(R \subseteq S \) is integral and has conductor \(C\), then \(R \subseteq S \) satisfies FCP if and only if \(S\) is a finitely generated \(R\)-module such that \(R/C\) is an Artinian ring. Among the motivations of their study, the authors mention the fact that the characterizations of FIP and FCP for integral extensions feature natural roles for the intermediate rings arising from seminormalization and t-closure.
reviewed by: Marco Fontana
Mathematics Subject Classification ID: 13B99
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13B21
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13G05
zbMATH DE Number: 6166665
zbMATH Keywords: normal pair
zbMATH Keywords: intermediare ring
zbMATH Keywords: chain
zbMATH Keywords: minimal ring extension
zbMATH Keywords: FIP
zbMATH Keywords: FCP
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2012.07.055
OpenAlex ID: W2074361033
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DOI: 10.1016/J.JALGEBRA.2012.07.055
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description: scientific article; zbMATH DE number 6166665