On \(\sigma\)-classes of modules with applications

In this paper we introduce some lattices of classes of left R-module relative to a preradical sigma. These lattices are generalizations of the lattices R-TORS, R-tors, R-nat, R-conat, of torsion theories, hereditary torsion theories, natural classes and conatural classes, respectively. We define the lattices \(\sigma\)-(R-TORS), \(\sigma\)-(R-tors), \(\sigma\)-(R-nat), \(\sigma\)-(R-conat), which reduce to the lattices mentioned above, when one takes sigma as the identity. We characterize the equality between these lattices by means of the (\(\sigma\)-HH) condition, which we introduce. We also present some results about \(\sigma\)-retractable rings, \(\sigma\)-Max rings extending results about Mod-retractable rings and Max rings.