The 2-stage Euclidean algorithm and the restricted Nagata's pairwise algorithm
type: Item
instance of: scholarly article
title: The 2-stage Euclidean algorithm and the restricted Nagata's pairwise algorithm
zbMATH Open document ID: 1239.13028
author: Ching-An Chen
author: Ming-Guang Leu
published in: Journal of Algebra
publication date: 2012-04-30T00:00:00Z
review text: The paper deals with two generalizations of the Euclidean algorithm in a commutative ring \(R\): the \(k\)-stage Euclidean algorithm introduced by \textit{G. E. Cooke} [J. Reine Angew. Math. 282, 133--156 (1976; Zbl 0328.13013)] and the restricted Nagata pairwise algorithm (\(RNPA\)) [\textit{M.-G. Leu}, Osaka Math. J., 45, 807--818 (2008; Zbl 1152.13016)]. In the case when \(R\) is a unique factorization domain the authors present a construction of the smallest \(RNPA\) in \(R\) (Theorem 2.3). Then they generalize the algorithm of Cooke, replacing in its definition the set of positive integers by an arbitrary well-ordered set and develop its properties. This permits them to construct infinitely many integral domains which are \(\omega\)-stage Euclidean, but not \(2\)-stage Euclidean. This was an open problem posed by Cooke [loc. cit.].
reviewed by: Władysław Narkiewicz
Mathematics Subject Classification ID: 13F07
Mathematics Subject Classification ID: 11R04
zbMATH DE Number: 6029081
zbMATH Keywords: Euclidean algorithm
zbMATH Keywords: restricted Nagata pairwise algorithm
zbMATH Keywords: \(k\)-stage Euclidean algorithm
zbMATH Keywords: \(\omega\)-stage Euclidean algorithm
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2011.09.029
OpenAlex ID: W2048168251
cites work: Q5563439
cites work: Q4138827
cites work: On Nagata's pairwise algorithm
cites work: On the structure of the \(GL_ 2\) of a ring
cites work: A weakening of the euclidean property for integral domains and applications to algebraic number theory. I.
cites work: The restricted Nagata's pairwise algorithm and the Euclidean algorithm
cites work: The Euclidean algorithm
cites work: About Euclidean rings
DOI: 10.1016/J.JALGEBRA.2011.09.029
description: scientific article; zbMATH DE number 6029081