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 {R_0} , that is, every onesided ideal ≠ 0 \ne 0 of R 0 {R_0} contains an ideal ≠ 0 \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 {R_0} is strongly right bounded and has finite left socle.