An investigation of the Baer-Kaplansky property

In this paper, we construct a local artinian ring R with Jacobson radical W such that W2=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W<^>2=0$$\end{document}, Q=R/W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q=R/W$$\end{document} is commutative, dim(QW)=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(_QW)=1$$\end{document} and dim(WQ)=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(W_Q)=2$$\end{document}. Then we show that, for this ring R, the category of all right R-modules Mod-R is not a Baer-Kaplansky class by proving that the class of all indecomposable right R-modules (all finitely generated right R-modules) is not Baer-Kaplansky. Finally, we give an application on some module classes over this constructed ring R.