The lattice of pre-natural classes of modules
by
Yiqiang Zhou
Memorial University of Newfoundland

Let R be a ring. A class F of R-modules is a hereditary pretorsion class if F is closed under submodules, direct sums, factor modules and isomorphic copies. A class L of R-modules is a natural class if L is closed under submodules, direct sums, injective hulls and isomorphic copies. A class K of R-modules is called a pre-natural class if K is the intersection of a hereditary pretorsion class and a natural class. In this talk, we introduce and discuss a new complete lattice whose elements are the 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.

Date received: March 17, 1999