Rings Without a Middle Class from a Lattice-Theoretic Perspective

SARAÇ B., Lopez-Permouth S. R. , Zamora-Erazo S.

MEDITERRANEAN JOURNAL OF MATHEMATICS, vol.17, no.2, 2020 (Peer-Reviewed Journal) identifier

  • Publication Type: Article / Article
  • Volume: 17 Issue: 2
  • Publication Date: 2020
  • Doi Number: 10.1007/s00009-020-1477-9
  • Journal Indexes: Science Citation Index Expanded, Scopus, MathSciNet, zbMATH, DIALNET


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.