REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS, no.1, 2025 (SCI-Expanded)
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 := [GRAPHICS] 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).