Representations of Quivers over Artinian Rings

Given a unitary artinian ring R and a finite acyclic quiver Q, let Λ := RQ be the path ring of Q over R. Then Gorenstein-projective Λ-modules are exactly the separated monic representations of Q over R which satisfy the local Gorenstein-projective condition. We denote by smon(Q,R) the category of all finitely generated separated monic representations of Q over R, and Gp(Λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {G}p({\varLambda })$\end{document} the category of all finitely generated Gorenstein-projective Λ-modules. If R is a selfinjective ring, then Gp(Λ)=smon(Q,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {G}p({\varLambda })=\text {smon}{\it (Q, R) }$\end{document}. As an application, if R is a commutative uniserial selfinjective ring of length 2 (here it means that as a regular module R is uniserial with length 2), let 0 ≠ a ∈radR and R̄=R/radR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\bar {R}=R/\text {rad}\textit {R}$\end{document}, our main result says that there is a full functor H:Gp(Λ)→modR̄Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H: \mathcal {G}p({\varLambda }) \to \text {mod} {\bar {R}Q}$\end{document} which induces a bijection between the indecomposable non-projetive Gorenstein-projective Λ-modules and the indecomposable R̄Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \bar {R}Q$\end{document}-modules. Moreover, in this case, each Gorenstein-projective Λ-module is strongly Gorenstein-projective.