Topics in Torsion Theory

Torsion theory provides an umbrella under which many classical properties of rings and modules can be reformulated. The purpose of this book is to provide the reader with a quick introduction to torsion theory and to study selected properties of rings and modules in this setting. The material presented ranges from a torsion theoretical treatment of standard topics in ring and module theory to how heretofore untreated properties of rings and modules might be dealt with in this setting. The approach has been to develop the material so that classical results can be recovered by selecting an appropriate torsion theory. Simple modules, maximal submodules, the Jacobson radical and modules with chain conditions are investigated relative to a torsion theory. A relative form of Nakayama's lemma is given and a generalized Hopkins-Levitzki theorem is established. Injective and projective concepts are studied and a generalized Bear's condition for injective modules and a generalized Fuchs condition for quasi-injective modules are shown to hold. Flat modules and covers and hulls of modules are investigated and the concept of a relative (quasi-) projective cover is used to characterize those rings which are right perfect modulo their torsion ideal. Torsion free covers are also studied and results are given which generalize well known results on torsion free covers for modules (with usual torsion) over an integral domain. Finally, (semi-) primitive, simple and (semi-) prime, semisimple and primitive rings are investigated relative to a torsion theory and, in this setting, right primitive rings are linked to a form of density which is reminiscent of Jacobson density for right primitive rings.