Characterizing rings in terms of the extent of the injectivity and projectivity of their modules

Given a ring R, we define its right i-profile (resp. right p-profile) to be the collection of injectivity domains (resp. projectivity domains) of its right R-modules. We study the lattice theoretic properties of these profiles and consider ways in which properties of the profiles may determine the structure of rings and vice versa. We show that the i-profile is isomorphic to an interval of the lattice of linear filters of right ideals of R, and is therefore modular and coatomic. In particular, we give a practical characterization of the profile of a right artinian ring and offer an example of a ring without injective left middle class for with the same is not true on the right-hand side. We characterize the p-profile of a right perfect ring and show through an example that the right p-profile of a ring is not necessarily a set. In addition, we use our results to provide a characterization of a special class of QF-rings in which the injectivity and projectivity domains of all modules coincide. The study of rings in terms of their (i- or p-)profile was inspired by the study of rings with no right (i- or p-)middle class, initiated in recent papers by Er, López-Permouth and Sökmez, and by Holston, López-Permouth and Orhan-Ertaş.