On some Euclidean properties of matrix algebras

38 pages, some minor corrections.International audienceLet $\mathfrak{R}$ 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