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arXiv · 1111.2090

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

Sergio R. López-Permouth·José E. Simental
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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, López-Permouth and Sökmez, and by Holston, López-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.

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BibTeXRIS

Sergio R. López-Permouth, José E. Simental. 2012-02-22. Characterizing rings in terms of the extent of injectivity and projectivity of their modules. https://doi.org/10.1016/j.jalgebra.2012.04.005

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