Rings of equivalent dominant and codominant dimensions

Abstract

Continuing an earlier examination of the codominant dimension of rings and modules, a categorical characterization is given for rings of equivalent dominant and codominant dimensions. Specifically, the question is teduced to when RR and the left minimal injective cogenerator 11 can be used as test modules respectively for the dominant dimension of the projective modules and the codominant dimension of the injective modules. These conditions are in turn characterized by when the injective projective modules are 2-injective. Also, a new and shortened version is given for the proof that the codominant dimension of the injective modules is equal to the dominant dimension of the projective modules.