On the Multiplicative Monoid of n×n Matrices Over Artinian Chain Rings

Abstract

Let R be an Artinian chain ring with a principal maximal ideal. We investigate properties of matrices over R and give matrix representations of R-submodules of R n first, then consider Green's relations, Green's relation equivalent classes, Schützenberger groups of 𝒟-classes, principal factors, and group ℋ-classes of the multiplicative monoid M n (R) of n × n matrices over R. Furthermore, we show that M n (R) is an eventually regular semigroup and derive basic numerical information of M n (R) when R is finite.