A ring theoretic approach to the finite representation type

An Artin algebra $\mathrm{\Lambda }$ is said to be of finite Cohen-Macaulay type, $\mathrm{C}\mathrm{M}$-finite for short, if the full subcategory $\mathrm{G}\mathrm{p}\mathrm{r}\mathrm{j}-\mathrm{\Lambda }$ of finitely generated Gorenstein projective $\mathrm{\Lambda }$-modules is of finite representation type. If $\mathrm{\Lambda }$ is a $\mathrm{C}\mathrm{M}$-finite algebra, then we denote by $\mathrm{A}\mathrm{u}\mathrm{s}(\underset{‾}{\mathrm{G}\mathrm{p}\mathrm{r}\mathrm{j}}-\mathrm{\Lambda })$ the stable Cohen-Macaulay Auslander algebra, i.e. ${\underset{‾}{\mathrm{E}\mathrm{n}\mathrm{d}}}_{\mathrm{\Lambda }}(G)$, where $G$ is a basic representation generator of $\mathrm{G}\mathrm{p}\mathrm{r}\mathrm{j}-\mathrm{\Lambda }$. In this paper, we will explain how by defining an equivalence relation on the elements of algebra $\mathrm{A}\mathrm{u}\mathrm{s}(\underset{‾}{\mathrm{G}\mathrm{p}\mathrm{r}\mathrm{j}}-\mathrm{\Lambda })$ can be used to give a characterization for $\mathrm{A}\mathrm{u}\mathrm{s}(\underset{‾}{\mathrm{G}\mathrm{p}\mathrm{r}\mathrm{j}}-\mathrm{\Lambda })$ to be of finite representation type, or equivalently, the $\mathrm{C}\mathrm{M}$-finiteness of the algebra of $2\times 2$ lower triangular matrices over $\mathrm{\Lambda },$ where $\mathrm{\Lambda }$ is a $\mathrm{C}\mathrm{M}$-finite Artin algebra over an algebraic closed filed. Then, by presenting some examples we will show how our results work.