New bad lines in \(R[x,y]\) and optimization of the epimorphism theorem
type: Item
instance of: scholarly article
title: New bad lines in \(R[x,y]\) and optimization of the epimorphism theorem
zbMATH Open document ID: 1107.14051
author: Stéphane Vénéreau
published in: Journal of Algebra
publication date: 2006-12-07T00:00:00Z
review text: The article under review studies conditions on a ring \(R\) under which all lines in \(R[x,y]\) are variables. A polynomial \(f(x,y)\in R[x,y]\) is called a line or an embedded line if \(R[x,y]/(f)\cong R[t]\). It is called a variable or a coordinate if there exists a \(g\in R[x,y]\) so that \(R[f,g]=R[x,y]\). If \(R\) is a field of characteristic zero, all lines are variables was proved by \textit{S. S. Abhyankar} and \textit{T. Moh} [J. Reine Angew. Math. 276, 148--166 (1975; Zbl 0332.14004)]. The characteristic zero hypothesis is essential and there are positive characteristic counterexamples due to \textit{M. Nagata} [J. Math. Kyoto Univ. 11, 149--154 (1971; Zbl 0213.23201)]. It was known that there are examples of \(R\) over which lines are not variables even if characteristic is zero. In this paper the author gives necessary and sufficient conditions on \(R\) under which any line in \(R[x,y]\) is a variable. The result involves a generalization of seminormality, which the author calls \(n\)-seminormality.
Mathematics Subject Classification ID: 14R10
Mathematics Subject Classification ID: 13F45
Mathematics Subject Classification ID: 14R25
zbMATH DE Number: 5078145
zbMATH Keywords: epimorphism theorem
zbMATH Keywords: embedded lines
zbMATH Keywords: variables
zbMATH Keywords: seminormality
reviewed by: N. Mohan Kumar
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.03.035
OpenAlex ID: W2128918934
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DOI: 10.1016/J.JALGEBRA.2006.03.035
description: scientific article; zbMATH DE number 5078145