Artinian Modules Over a Matrix Ring

Abstract

Let $\mathbb{Z},\mathbb{Q}$ be the ring of integers and the field of rational numbers. Denote by Λ the matrix ring

$$ \left( {\begin{array}{*{20}{c}} \mathbb{Q}&0 \\ \mathbb{Q}&\mathbb{Z} \end{array}} \right) = \left\{ {\left. {\left( {\begin{array}{*{20}{c}} \alpha &0 \\ \beta &m \end{array}} \right) } \right|\alpha ,\beta \in \mathbb{Q},m \in \mathbb{Z}} \right\} $$