LEZOWSKI, Pierre

doi:10.1016/j.jalgebra.2017.05.018

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LEZOWSKI, Pierre
Laboratoire de Mathématiques Blaise Pascal [LMBP]
Lithe and fast algorithmic number theory [LFANT]

Langue

Article de revue

Ce document a été publié dans

Journal of Algebra. 2017-09-15, vol. 486, p. 157--203

Elsevier

Résumé en anglais

Let $\mathfrak{R}$ be a commutative ring and $n \in \mathbf{Z}_{>1}$. We study some Euclidean properties of the algebra $\mathrm{M}_{n}(\mathfrak{R})$ of $n$ by $n$ matrices with coefficients in $\mathfrak{R}$. In particular, we prove that $\mathrm{M}_{n}(\mathfrak{R})$ is a left and right Euclidean ring if and only if $\mathfrak{R}$ is a principal ideal ring. We also study the Euclidean order type of $\mathrm{M}_{n}(\mathfrak{R})$. If $\mathfrak{R}$ is a K-Hermite ring, then $\mathrm{M}_{n}(\mathfrak{R})$ is a $(4n-3)$-stage left and right Euclidean. We obtain shorter division chains when $\mathfrak{R}$ is an elementary divisor ring, and even shorter ones when $\mathfrak{R}$ is a principal ideal ring. If we assume that $\mathfrak{R}$ is an integral domain, $\mathfrak{R}$ is a Bézout ring if and only if $\mathrm{M}_{n}(\mathfrak{R})$ is $\omega$-stage left and right Euclidean.< Réduire

Mots clés en anglais

K-Hermite ring

Euclidean ring

division chains

elementary divisor ring

2-stage Euclidean ring

Principal Ideal Ring

URI

https://oskar-bordeaux.fr/handle/20.500.12278/193639

DOI

http://dx.doi.org/10.1016/j.jalgebra.2017.05.018

Projet Européen

Algorithmic Number Theory in Computer Science

Origine

Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251