On some Euclidean properties of matrix algebras

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.
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Pierre Lezowski. On some Euclidean properties of matrix algebras. Journal of Algebra, Elsevier, 2017, 486, pp.157--203. ⟨10.1016/j.jalgebra.2017.05.018⟩. ⟨hal-01135202v3⟩

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