ABSTRACT
Section 2.1 outlines the main facts about torsion theories. Proofs are not given in section 2.1, because it is intended merely for giving definitions. Section 2.2 develops the basic properties of the category σ[M ] subgenerated by an arbitrary module M . It includes the Goldie torsion class in σ[M ]. The next section 2.3 covers natural classes, which is a central topic here. Then section 2.4 is about M -natural classes, which can be regarded as natural classes relativized to the subcategory σ[M ] of Mod-R. The last section 2.5 on prenatural classes gives a generalization of natural classes. A natural class, or an M -natural class, is a special case of a pre-natural class. Their importance and usefulness will not become fully apparent until Chapter 6, where it will be shown that almost all lattices of module classes associated to a ring are sublattices of the lattice of pre-natural classes.
