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Given a ring R, we define its right i-profile (resp. right p-profile) to bethe collection of injectivity domains (resp. projectivity domains) of its rightR-modules. We study the lattice theoretic properties of these profiles andconsider ways in which properties of the profiles may determine the structureof rings and viceversa. We show that the i-profile is isomorphic to an intervalof the lattice of linear filters of right ideals of R, and is therefore modularand coatomic. In particular, we give a practical characterization of thei-profile of a right artinian ring. We show through an example that thep-profile is not necessarily a set, and also characterize the right p-profileof a right perfect ring. The study of rings in terms of their (i- or p-)profilewas inspired by the study of rings with no (i- or p-) middle class, initiatedin recent papers by Er, L\'opez-Permouth and S\"okmez, and by Holston,L\'opez-Permouth and Orhan-Ertas. In this paper, we obtain further resultsabout these rings and we also use our results to provide a characterization ofa special class of QF-rings in which the injectivity and projectivity domainsof any module coincide.

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