On the Baer–Kaplansky theorem for injective modules

In this paper, we study the Baer–Kaplansky theorem for injective modules. Firstly, we prove that every right semi-artinian local ring satisfies the Baer–Kaplansky theorem for injective modules. Later, we work on the commutative principal ideal domains. We prove that a commutative local principal ideal domain (i.e. discrete valuation ring) satisfies the Baer–Kaplansky theorem for completely virtually semisimple modules. Finally, we make examples on the upper triangular matrix ring \(A:=\left[ \begin{matrix} R & M \\ 0 & S \end{matrix}\right] \) $A:=\left[\begin{array}{cc}R& M\\ 0& S\end{array}\right]$ with nonzero M showing that the Baer–Kaplansky theorem fails for injective right A-modules, even if R and S are semisimple rings. We deduce that for every division ring D and \(n > 1\) $n>1$ , the Baer–Kaplansky theorem fails for injective right modules over the ring \(T_n(D)\) $T_n(D)$ (upper triangular matrix ring over D).