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

TL;DR

This paper introduces and analyzes the concepts of i-profile and p-profile of rings, exploring their lattice properties and how they relate to ring structure, with applications to rings with no middle class and QF-rings.

Contribution

It characterizes the i-profile and p-profile of rings, relates them to lattice structures, and applies these concepts to classify certain classes of rings.

Findings

i-profile is isomorphic to an interval of the lattice of linear filters of right ideals

The i-profile of a right artinian ring can be practically characterized

The p-profile may not be a set, and its structure is characterized for right perfect rings

Abstract

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.