On lattices associated to rings with respect to a preradical

. We introduce some new lattices of classes of modules with respect to appropriate preradicals. We introduce some concepts associated with these lattices, such as the σ -semiartinian rings, the σ -retractable modules, the σ - V -rings, the σ -max rings. We continue to study σ -torsion theories, σ -open classes, σ -stable classes. We prove some theorems that extend some known results. Our results fall into well known situations when the preradical σ is chosen as the identity preradical.