On some Euclidean properties of matrix algebras

Let R be a commutative ring and n∈Z>1. We study some Euclidean properties of the algebra Mn(R) of n by n matrices with coefficients in R. In particular, we prove that Mn(R) is a left and right Euclidean ring if and only if R is a principal ideal ring. We also study the Euclidean order type of Mn(R). If R is a K-Hermite ring, then Mn(R) is (4n−3)-stage left and right Euclidean. We obtain shorter division chains when R is an elementary divisor ring, and even shorter ones when R is a principal ideal ring. If we assume that R is an integral domain, R is a Bézout ring if and only if Mn(R) is ω-stage left and right Euclidean.