On the Baer–Kaplansky theorem for injective modules: On the Baer–Kaplansky

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 : = R M 0 S 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 , the Baer–Kaplansky theorem fails for injective right modules over the ring T n ( D ) (upper triangular matrix ring over D ).