Rings whose cyclic modules have restricted injectivity domains

A poor module is one that is injective relative only to semisimple modules and a module is maximally injective if its domain of injectivity is a coatom in the lattice of domains of injectivity (the so called injective profile of a ring). A ring is said to have no right middle class if every right module is injective or poor. We consider two families of rings characterized by their cyclic right modules: those whose nonzero cyclic modules are poor (property (P)) and those whose nonzero cyclic modules are maximally injective (property (Q)). We show that rings satisfying property (Q) are precisely those rings that have no right middle class and property (P). Structural properties of both classes of rings are obtained and it is shown, in particular, that a ring with property (Q) is isomorphic to a matrix ring over a nonsemisimple local right Artinian ring and is such that its right socle, right singular ideal and Jacobson radical coincide.