Rings whose modules have maximal or minimal injectivity domains

In a recent paper, Alahmadi, Alkan and López–Permouth defined a module M to be poor if M is injective relative only to semisimple modules, and a ring to have no right middle class if every right module is poor or injective. We prove that every ring has a poor module, and characterize rings with semisimple poor modules. Next, a ring with no right middle class is proved to be the ring direct sum of a semisimple Artinian ring and a ring T which is either zero or of one of the following types: (i) Morita equivalent to a right PCI-domain, (ii) an indecomposable right SI-ring which is either right Artinian or a right V-ring, and such that soc(TT) is homogeneous and essential in TT and T has a unique simple singular right module, or (iii) an indecomposable right Artinian ring with homogeneous right socle coinciding with the Jacobson radical and the right singular ideal, and with unique non-injective simple right module. In case (iii) either TT is poor or T is a QF-ring with J2(T)=0. Converses of these cases are discussed. It is shown, in particular, that a QF-ring R with J2(R)=0 and homogeneous right socle has no middle class.