Synopses & Reviews
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.
About the Author
S. K. Jain is a Distinguished Professor Emeritus, Ohio University and Advisor, 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 also on the editorial board of the Electronic Journal of Algebra.
Ashish Srivastava is an Assistant Professor of Mathematics at Saint Louis University, Saint Louis, USA. He has written 15 research articles in Noncommutative Algebra and Combinatorics that have been published in various journals.
Askar A. Tuganbaev is a Professor of Mathematics at the Russian State University of Trade and Economics, Moscow, Russia. He has written 10 monographs and more than 180 research articles in Algebra that have been published in various journals.
Table of Contents
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