ABSTRACT
Natural classes and pre-natural classes provide a nice setting for study of chain conditions of rings and modules. For a fixed pre-natural class K of right R-modules and a fixed right R-module M , the ascending chain condition and descending chain condition on HK(M) are characterized in terms of certain decomposition properties of modules in K. This gives a unifying treatment of chain conditions of rings and modules in torsion theory and in the category σ[M ], because of the fact that all hereditary torsion classes, hereditary torsion free classes, and σ[M ] are pre-natural classes. When K is specialized to some well-selected pre-natural classes that are not of the previous three kinds, then new results are obtained by this approach.
