Pointwise minimal extensions
type: Item
instance of: scholarly article
title: Pointwise minimal extensions
zbMATH Open document ID: 1402.13009
author: Paul-Jean Cahen
author: Gabriel Picavet
author: Martine Picavet-L'Hermitte
published in: Arabian Journal of Mathematics
publication date: 2018-12-05T00:00:00Z
review text: The notion of minimal extension of rings was introduced by \textit{D. Ferrand} et al. [J. Algebra. 16, No. 3, 461--471 (1970; Zbl 0218.13011)]. \textit{P.-J. Cohen} et al. [J. Algebra Appl. 9, No. 1, 43--72 (2010; Zbl 1191.13009)] introduced pointwise minimal extensions, in the special context of domains. An extension \(R\subset S\) is said to be pointwise minimal extension if every simple sub-extension \(R\subset R[t]\) is minimal. The minimal pointwise extension \(R\subset S\) is called pointwise minimal pair if, for each \(T\) between \(R\) and \(S\), \(T\subset S\) is a pointwise minimal extension. The main goal of the paper under review is to characterized pointwise minimal extensions and pointwise minimal pair. The authors show first that pointwise minimal extension is either integral or integral closed. In Section 4, they focus on extensions over a field \(k\) and prove that there are four types of minimal extensions: decomposed, inert, ramified, and composite. In Section 5, the authors turn back to general integral extensions and fully characterize each type of pointwise minimal extensions (decomposed, inert, ramified, and composite). Numerous examples are contained in the last section of the paper.
reviewed by: Nikolay I. Kryuchkov
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 13B21
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13B30
zbMATH DE Number: 6989342
zbMATH Keywords: pointwise minimal extension
zbMATH Keywords: pointwise minimal pair extension
zbMATH Keywords: integral extension
zbMATH Keywords: integral closed extension
zbMATH Keywords: local ring
MaRDI profile type: Publication
full work available at URL: s40065-018-0202-z
OpenAlex ID: W2963291413
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Wikidata QID: Q130118929
DOI: 10.1007/S40065-018-0202-Z
description: scientific article; zbMATH DE number 6989342