Cyclic Modules and the Structure of Rings

Hardcover | December 15, 2012

byS.K. Jain, Ashish K. Srivastava, Askar A. Tuganbaev

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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 homologicalproperties 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 thebenefit 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.

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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 homologicalproperties are closely interrelated in the sense th...

S. K. Jain is a Distinguished Professor Emeritus at Ohio University and Advisor at King Abdulaziz University. He was at the Department of Mathematics at Ohio University from 1970-2009. He is an Executive Editor of the Journal of Algebra and its Applications (World Scientific) and Bulletin of Mathematical Sciences (Springer). He is al...

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Format:HardcoverDimensions:296 pages, 9.21 × 6.14 × 0.1 inPublished:December 15, 2012Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:019966451X

ISBN - 13:9780199664511

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Table of Contents

Preface1. Preliminaries2. Rings characterized by their proper factor rings3. Rings each of whose proper cyclic modules has a chain condition4. Rings each of whose cyclic modules is injective (or CS)5. Rings each of whose proper cyclic modules is injective6. Rings each of whose simple modules is injective (or -injective)7. Rings each of whose (proper) cyclic modules is quasi-injective8. Rings each of whose (proper) cyclic modules is continuous9. Rings each of whose (proper) cyclic modules is pi-injective10. Rings with cyclics @0-injective, weakly injective or quasi-projective11. Hypercyclic, q-hypercyclic and pi-hypercyclic rings12. Cyclic modules essentially embeddable in free modules13. Serial and distributive modules14. Rings characterized by decompositions of their cyclic modules15. Rings each of whose modules is a direct sum of cyclic modules16. Rings each of whose modules is an I0-module17. Completely integrally closed modules and rings18. Rings each of whose cyclic modules is completely integrally closed19. Rings characterized by their one-sided idealsReferencesIndex