An investigation of the Baer–Kaplansky property

In this paper, we construct a local artinian ring R with Jacobson radical W such that \(W^2=0\) $W^2=0$ , \(Q=R/W\) $Q=R/W$ is commutative, dim \((_QW)=1\) ${(}_{Q}W)=1$ and dim \((W_Q)=2\) $(W_Q)=2$ . 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.