Injective cogenerator rings and a theorem of Tachikawa. II

Abstract:

The main theorem states that a right injective cogenerator ring R has strongly bounded basic ring ${R_0}$, that is, every onesided ideal $\ne 0$ of ${R_0}$ contains an ideal $\ne 0$. A right injective cogenerator ring R is characterized by the condition: (1) R is semiperfect and right self-injective and (2) R has (finite) essential right socle. We show that (2) can be replaced by (2’) ${R_0}$ is strongly right bounded and has finite left socle.