Abstract
We further the study of rings with no middle class by focusing on an interpretation of that property in terms of the lattice of hereditary pretorsion classes over a given ring. For non-semisimple rings, the absence of a middle class is equivalent to the requirement that the class of all semisimple right modules be a coatom in that lattice. Taking advantage of this perspective, we discover new facts and shed light on others already known with a possibly more direct interpretation without having to refer to an exhaustive analysis of the structure theorems available in the literature. Our approach also allows us to characterize rings with no middle class in terms of hereditary pretorsion classes containing the class of all singular right modules. We discuss the open problem of whether there is a ring with no right middle class which is not right Noetherian and see, in particular, that an indecomposable ring satisfying that property would have to be Morita equivalent to a certain type of subring of a full linear ring.
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Acknowledgements
A part of this work was done during the visit of the first and the third authors to Ohio University, Center of Ring Theory and its Applications. The first author’s visit was supported by Hacettepe University Research Grant (FBI-2016-10374) and the third author’s visit was supported by Beca Mixta de Movilidad en el Extranjero scholarship from CONACyT Mexico. The first author also gratefully acknowledges the support he has received from the Turkish Scientific Research Council (TÜBİTAK) with Grant No. 117F084. The authors also would like to express their gratitude to the anonymous referee for his/her comments which improved the presentation of this paper.
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Saraç, B., López-Permouth, S.R. & Zamora-Erazo, S. Rings Without a Middle Class from a Lattice-Theoretic Perspective. Mediterr. J. Math. 17, 55 (2020). https://doi.org/10.1007/s00009-020-1477-9
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DOI: https://doi.org/10.1007/s00009-020-1477-9