Big lattices of module classes induced by preradicals

Abstract

We introduce new big lattices of classes of R-modules induced by suitable preradicals, the big lattice of $$\sigma $$-hereditary classes, the big lattice of$$\sigma $$-cohereditary classes, and the big lattice of $$\sigma $$-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$$\sigma $$-hereditary torsion theories, for a exact preradical $$\sigma $$. In case that $$\sigma $$ be the identity preradical, we obtain the usual lattice of hereditary torsion classes.