On some Euclidean properties of matrix algebras
Résumé
Let $\mathfrak{R}$ be a commutative ring and $n\in \mathbf{Z}_{>1}$. We study some Euclidean properties of the matrix algebra $M_n(\mathfrak{R})$ of $n$ by $n$ matrices with coefficients in $\mathfrak{R}$. In particular, we prove that $M_n(\mathfrak{R})$ 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 $M_n(\mathfrak{R})$. If $\mathfrak{R}$ is a principal ideal ring, we prove that $M_n(\mathfrak{R})$ is $2$-stage Euclidean and every pair of matrices in $M_n(\mathfrak{R})$ admits a terminating division chain whose length is at most $3n-1$.
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