When only finitely many intermediate rings result from juxtaposing two minimal ring extensions
type: Item
instance of: scholarly article
zbMATH Open document ID: 1354.13015
author: David E. Dobbs
author: Jay Shapiro
publication date: 2016-10-06T00:00:00Z
full work available at URL: 2 DobbsPJM.pdf
Mathematics Subject Classification ID: 13B99
Mathematics Subject Classification ID: 13B21
Mathematics Subject Classification ID: 13G05
zbMATH DE Number: 6633367
zbMATH Keywords: commutative ring
zbMATH Keywords: minimal ring extension
zbMATH Keywords: FIP
zbMATH Keywords: integral
zbMATH Keywords: integrally closed
zbMATH Keywords: ramified
zbMATH Keywords: inert
zbMATH Keywords: decomposed
zbMATH Keywords: FCP
zbMATH Keywords: seminormalization
zbMATH Keywords: integral domain
MaRDI profile type: Publication
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title: When only finitely many intermediate rings result from juxtaposing two minimal ring extensions
published in: Palestine Journal of Mathematics
review text: A ring extension \(A\subsetneq B\) of commutative rings is a \textit{minimal extension} of \(A\) if there is no ring \(C\) such \(A\subsetneq C \subsetneq B\). Clearly, a minimal ring extension is either integrally closed or integral. For rings \(R\subsetneq S\subsetneq T\) such that \(R\subsetneq S\) and \(S\subsetneq T\) are minimal ring extensions, the authors present necessary and sufficient conditions for the extension \(R\subsetneq T\) to have only finitely many intermediate rings (condition FIP). For example, by a previous result of the present authors, if the extension \(S\subsetneq T\) is integraly closed, then the extension \(R\subsetneq T\) satisfies FIP. To characterize the FIP property of the extension \(R\subsetneq T\) the authors take into account the possible three properties of an integral minimal extension (inert, ramified and decomposed. The definitions are recalled in the paper.) Examples are provided for a full analysis of all the possibilities. Moreover, using this analysis, the authors study the case when the extension \(R\subsetneq T\) contains just one proper intermediate ring. This study is completed by the first author in the paper [Palest. J. Math. 6, No. 1, 31--44 (2017; Zbl 1354.13014)].
reviewed by: Moshe Roitman
description: scientific article; zbMATH DE number 6633367