Some Results on SI-Rings Original Research Article

The following results on a ring R are obtained: (i) R is the ring direct sum of a semiprimary SI-ring and a rignt CS right SI-ring with zero right socle if and only if every cyclic semiprimitive right R-module is a direct sum of a projective module and an injective module; (ii) R is the ring direct sum of a semiprimary SI-ring and an SI-ring with zero right (and left) socle if and only if every finitely (or 2-) generated semiprimitive right R-module is a direct sum of a projective module and an injective module; (iii) R is the ring direct sum of a semisimple ring and a right SI-domain if and only if every cyclic semiprimitive right R-module is projective or injective, as a consequence, R is semisimple if and only if every 2-generated semiprimitive right R-module is projective or injective.