## Abstract

As with Euclidean rings and rings admitting a restricted Nagata's pairwise algorithm, we will give an internal characterization of 2-stage Euclidean rings. Applying this characterization we are capable of providing infinitely many integral domains which are ω-stage Euclidean but not 2-stage Euclidean. Our examples solve finally a fundamental question related to the notion of k-stage Euclidean rings raised by G.E. Cooke [G.E. Cooke, A weakening of the Euclidean property for integral domains and applications to algebraic number theory I, J. Reine Angew. Math. 282 (1976) 133-156]. The question was stated as follows: "I do not know of an example of an ω-stage euclidean ring which is not 2-stage euclidean.". Also, in this article we will give a method to construct the smallest restricted Nagata's pairwise algorithm θ on a unique factorization domain which admits a restricted Nagata's pairwise algorithm. It is of interest to point out that in a Euclidean domain the shortest length d(a, b) of all terminating division chains starting from a pair (a, b) and the value θ(a, b) with g.c.d.(a, b) ≠ 1 can be determined by each other.

Original language | English |
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Pages (from-to) | 1-13 |

Number of pages | 13 |

Journal | Journal of Algebra |

Volume | 348 |

Issue number | 1 |

DOIs | |

State | Published - 15 Dec 2011 |

## Keywords

- Euclidean algorithm
- K-stage Euclidean algorithm
- Primary
- Restricted Nagata's pairwise algorithm
- Secondary
- ω-stage Euclidean algorithm