We introduce new big lattices of classes of R-modules induced by suitable preradicals, the big lattice of σ-hereditary classes, the big lattice of σ-cohereditary classes, and the big lattice of σ-open classes. We prove that their Skeletons are boolean lattices which generalize and extend the lattice of natural classes, the lattice of conatural classes and the Skeleton of the lattice of hereditary torsion theories, respectively. We also introduce the lattice of σ-hereditary torsion theories, for a exact preradical σ. In case that σ be the identity preradical, we obtain the usual lattice of hereditary torsion classes.
Some properties of and relations between several (big) lattices of module classes are used in this paper to obtain information about the ring over which modules are taken. The authors reach characterizations of trivial rings, semisimple rings and certain rings over which every torsion theory is hereditary.
This article consists of two sections. In the first one, the concepts of spanning and cospanning classes of modules, both hereditarily and cohereditarily, are explained, and some closure properties of the class of modules hereditarily cospanned by a conatural class are established, which amount to its being a hereditary torsion class. This gives a function from R-conat to R-tors and it is proven that its being a lattice isomorphism is part of a characterization of bilaterally perfect rings. The second section begins considering a description of pseudocomplements in certain lattices of module classes. The idea is generalized to define an inclusion-reversing operation on the collection of classes of modules. Restricted to R-nat, it is shown to be a function onto R-tors, and its being an anti-isomorphism is equivalent to R being left semiartinian. Lastly, another characterization of R being left semiartinian is given, in terms solely of R-tors.
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