نشریه علوم ریاضی و انفورماتیک

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 σ-(R-TORS), σ-(R-tors), σ-(R-nat), σ-(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 (σ-HH) condition, which we introduce. We also present some results about σ-retractable rings, σ-Max rings extending results about Mod-retractable rings and Max rings.