The 2-stage Euclidean algorithm and the restricted Nagata's pairwise algorithm
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.].
scientific article; zbMATH DE number 6029081