On finiteness of chains of intermediate rings
type: Item
instance of: scholarly article
title: On finiteness of chains of intermediate rings
zbMATH Open document ID: 1180.13008
author: M. Ben Nasr
published in: Monatshefte für Mathematik
publication date: 2009-09-09T00:00:00Z
review text: Let \(R\subseteq S\) be an extension of integral domains. This extension is said to have the ``finite length of intermediate chains of domains'' property (FICP) if each chain of intermediate rings between \(R\) and \(S\) is finite. Let \(T\) be a ring between \(R\) and \(S\). Certainly if \(R\subseteq S\) has FICP, so do the extensions \(R\subseteq T\) and \(T\subseteq S\). The converse is false as \( \mathbb{Z}\subseteq \mathbb{Z}_{(p)}\) and \(\mathbb{Z}_{(p)}\subseteq \mathbb{ Q}\) both have FICP for any nonzero prime \(p\), but \(\mathbb{Z}\subseteq \mathbb{Q}\) certainly does not. Now \textit{R. Gilmer} [Proc. Am. Math. Soc. 131, No. 8, 2337--2346 (2003; Zbl 1017.13009)] showed that for \(D\) an integral domain with quotient field \(K\), \(D\subseteq K\) has FICP if and only if \(D\subseteq D^{\prime }\) and \(D^{\prime }\subseteq K\) have FICP (where \(D^{\prime }\) is the integral closure of \(D\)\ in \(K\)) if and only if \(D^{\prime }\) is a Prüfer domain with a finite number of prime ideals. The main result of the paper under review is the following generalization of Gilmer's result. Theorem: Let \( R\subseteq S\) be an extension of integral domains and let \(R^{\ast }\) be the integral closure of \(R\) in \(S\). Suppose that \(R^{\ast }\) is finite dimensional semiquasilocal. If \(R\subseteq R^{\ast }\) and \(R^{\ast }\subseteq S\) have FICP, then so does \(R\subseteq S\). An example using pullbacks is given where \(R\subset S\) has FICP, but \(R^{\ast }\) is not a Prüfer domain.
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13G05
zbMATH DE Number: 5602534
zbMATH Keywords: ring extension
zbMATH Keywords: intermediate chain of domains
zbMATH Keywords: normal pair
zbMATH Keywords: valuation domain
reviewed by: D. D. Anderson
MaRDI profile type: Publication
full work available at URL: s00605-008-0090-y
OpenAlex ID: W2071978702
cites work: The primitive element theorem for commutative algebras
cites work: Residually algebraic pairs of rings
cites work: Couples d'anneaux partageant un idéal. (Couples of rings sharing an ideal)
cites work: Overrings of Commutative Rings. III: Normal Pairs
cites work: Some finiteness conditions on the set of overrings of an integral domain
cites work: Finiteness of the set of intermediary rings in normal pairs
cites work: Q4049183
DOI: 10.1007/S00605-008-0090-Y
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description: scientific article; zbMATH DE number 5602534