Cyclic Modules and the Structure of Rings

This unique and comprehensive volume provides an up-to-date account of the literature on the subject of determining the structure of rings over which cyclic modules or proper cyclic modules have a finiteness condition or a homological property. The finiteness conditions and homological properties are closely interrelated in the sense that either hypothesis induces the other in some form. This is the first book to bring all of this important material on the subject together. Over the last 25 years or more numerous mathematicians have investigated rings whose factor rings or factor modules have a finiteness condition or a homological property. They made important contributions leading to new directions and questions, which are listed at the end of each chapter for the benefit of future researchers. There is a wealth of material on the topic which is combined in this book, it contains more than 200 references and is not claimed to be exhaustive. This book will appeal to graduate students, researchers, and professionals in algebra with a knowledge of basic noncommutative ring theory, as well as module theory and homological algebra, equivalent to a one-year graduate course in the theory of rings and modules.Preface ; 1. Preliminaries ; 2. Rings characterized by their proper factor rings ; 3. Rings each of whose proper cyclic modules has a chain condition ; 4. Rings each of whose cyclic modules is injective (or CS) ; 5. Rings each of whose proper cyclic modules is injective ; 6. Rings each of whose simple modules is injective (or -injective) ; 7. Rings each of whose (proper) cyclic modules is quasi-injective ; 8. Rings each of whose (proper) cyclic modules is continuous ; 9. Rings each of whose (proper) cyclic modules is pi-injective ; 10. Rings with cyclics @0-injective, weakly injective or quasi-projective ; 11. Hypercyclic, q-hypercyclic and pi-hypercyclic rings ; 12. Cyclic modules essentially embeddable in free modules ; 13. Serial and distributive modules ; 14. Rings characterized by decompositions of their cyclic modules ; 15. Rings each of whose modules is a direct sum of cyclic modules ; 16. Rings each of whose modules is an I0-module ; 17. Completely integrally closed modules and rings ; 18. Rings each of whose cyclic modules is completely integrally closed ; 19. Rings characterized by their one-sided ideals ; References ; Index