On some Euclidean properties of matrix algebras

Pierre Lezowski 1, 2
1 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : 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.
Document type :
Preprints, Working Papers, ...
Liste complète des métadonnées

https://hal.archives-ouvertes.fr/hal-01135202
Contributor : Pierre Lezowski <>
Submitted on : Monday, April 27, 2015 - 4:06:31 PM
Last modification on : Thursday, January 11, 2018 - 6:22:36 AM
Document(s) archivé(s) le : Monday, September 14, 2015 - 2:06:13 PM

Files

matrices.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-01135202, version 2

Collections

Citation

Pierre Lezowski. On some Euclidean properties of matrix algebras. 2015. ⟨hal-01135202v2⟩

Share

Metrics

Record views

270

Files downloads

167