The lattice of pre-natural classes of modules

In this paper, we introduce and study a new complete lattice whose elements are the so-called pre-natural classes of R-modules. This lattice 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. Moreover, the complete Boolean lattice of all natural classes of R-modules is a sublattice of this lattice. Various properties of this new lattice are formulated and some applications to the ring R are given. (C) 1999 Elsevier Science B.V. All rights reserved.