Characterizing rings in terms of the extent of the injectivity and projectivity of their modules.
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Publication:1938538
DOI10.1016/J.JALGEBRA.2012.04.005zbMATH Open1284.16003arXiv1111.2090OpenAlexW2159433015WikidataQ122618595 ScholiaQ122618595MaRDI QIDQ1938538
Author name not available (Why is that?)
Publication date: 21 February 2013
Published in: (Search for Journal in Brave)
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.
Full work available at URL: https://arxiv.org/abs/1111.2090
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