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.