On some Euclidean properties of matrix algebras
hal.structure.identifier | Laboratoire de Mathématiques Blaise Pascal [LMBP] | |
hal.structure.identifier | Lithe and fast algorithmic number theory [LFANT] | |
dc.contributor.author | LEZOWSKI, Pierre | |
dc.date.accessioned | 2024-04-04T03:09:50Z | |
dc.date.available | 2024-04-04T03:09:50Z | |
dc.date.created | 2016-09-19 | |
dc.date.issued | 2017-09-15 | |
dc.identifier.issn | 0021-8693 | |
dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/193639 | |
dc.description.abstractEn | 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. | |
dc.language.iso | en | |
dc.publisher | Elsevier | |
dc.subject.en | K-Hermite ring | |
dc.subject.en | Euclidean ring | |
dc.subject.en | division chains | |
dc.subject.en | elementary divisor ring | |
dc.subject.en | 2-stage Euclidean ring | |
dc.subject.en | Principal Ideal Ring | |
dc.title.en | On some Euclidean properties of matrix algebras | |
dc.type | Article de revue | |
dc.identifier.doi | 10.1016/j.jalgebra.2017.05.018 | |
dc.subject.hal | Mathématiques [math]/Anneaux et algèbres [math.RA] | |
dc.subject.hal | Mathématiques [math]/Théorie des nombres [math.NT] | |
dc.description.sponsorshipEurope | Algorithmic Number Theory in Computer Science | |
bordeaux.journal | Journal of Algebra | |
bordeaux.page | 157--203 | |
bordeaux.volume | 486 | |
bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
bordeaux.institution | Université de Bordeaux | |
bordeaux.institution | Bordeaux INP | |
bordeaux.institution | CNRS | |
bordeaux.peerReviewed | oui | |
hal.identifier | hal-01135202 | |
hal.version | 1 | |
hal.popular | non | |
hal.audience | Internationale | |
hal.origin.link | https://hal.archives-ouvertes.fr//hal-01135202v1 | |
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