  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 viceversa. 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
i-profile of a right artinian ring. We show through an example that the
p-profile is not necessarily a set, and also characterize the right p-profile
of a right perfect ring. The study of rings in terms of their (i- or p-)profile
was inspired by the study of rings with no (i- or p-) middle class, initiated
in 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 results
about these rings and we also use our results to provide a characterization of
a special class of QF-rings in which the injectivity and projectivity domains
of any module coincide.
