ABSTRACT
The collectionN pr (R) of pre-natural classes of R-modules is a complete lattice, the study of which was initiated in [142]. It contains as subsets almost all the important lattices of module classes associated with the ring R. For example, N pr (R) contains a complete sublattice isomorphic to the complete lattice of all linear topologies of R and a sublattice anti-isomorphic to the frame of all hereditary torsion theories of R. The complete Boolean lattice of all natural classes of R-modules is also a sublattice of N pr (R). The lattice N pr (R) and some of its sublattices are introduced and discussed in sections 6.1 and 6.2. In section 6.3, several properties of the lattice N pr (R) and of some of its sub-
lattices are discussed and related to properties of the ring R and the category Mod-R. In section 6.4, the rings R for which every hereditary pretorsion class is hereditary torsion are investigated as applications. Section 6.5 explores the functoriality of Nr(·), and some of its subfunctors. Finally in section 6.6, it is shown that every ring R contains a lattice of ideals isomorphic to Nf (R), the lattice of all natural classes of nonsingular R-modules.
