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On some Euclidean properties of matrix algebras

hal.structure.identifierLaboratoire de Mathématiques Blaise Pascal [LMBP]
hal.structure.identifierLithe and fast algorithmic number theory [LFANT]
dc.contributor.authorLEZOWSKI, Pierre
dc.date.accessioned2024-04-04T03:09:50Z
dc.date.available2024-04-04T03:09:50Z
dc.date.created2016-09-19
dc.date.issued2017-09-15
dc.identifier.issn0021-8693
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/193639
dc.description.abstractEnLet $\mathfrak{R}$ 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.isoen
dc.publisherElsevier
dc.subject.enK-Hermite ring
dc.subject.enEuclidean ring
dc.subject.endivision chains
dc.subject.enelementary divisor ring
dc.subject.en2-stage Euclidean ring
dc.subject.enPrincipal Ideal Ring
dc.title.enOn some Euclidean properties of matrix algebras
dc.typeArticle de revue
dc.identifier.doi10.1016/j.jalgebra.2017.05.018
dc.subject.halMathématiques [math]/Anneaux et algèbres [math.RA]
dc.subject.halMathématiques [math]/Théorie des nombres [math.NT]
dc.description.sponsorshipEuropeAlgorithmic Number Theory in Computer Science
bordeaux.journalJournal of Algebra
bordeaux.page157--203
bordeaux.volume486
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-01135202
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-01135202v1
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