  In this paper a simple right R-module S over a ring R is called hypersimple
if its injective hull E(S) is cyclic, and a ring R is called right hypersimple
if every simple right R-module is hypersimple. We initiate a study of these new
notions, and revisit Osofsky's work on hypercyclic rings, i.e. rings whose
cyclic right modules have cyclic injective hulls.
