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Extensions of Rings and Modules by Gary F Birkenmeier, Jae Keol Park, S Tariq Rizvi

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About this book :-
Extensions of Rings and Modules written by Gary F Birkenmeier, Jae Keol Park, S Tariq Rizvi
Each section of the book is enriched with examples and exercises which make this monograph useful not only for experts but also as a text for advanced graduate courses. Historical notes appear at the end of each chapter, and a list of Open Problems and Questions is provided to stimulate further research in this area. With over 400 references, Extensions of Rings and Modules will be of interest to researchers in algebra and analysis and to advanced graduate students in mathematics.
Since the discovery of the existence of the injective hull of an arbitrary module independently in 1952 by Shoda and in 1953 by Eckmann and Schopf, there have been numerous papers dedicated to the study and description of various types of hulls or “minimal” extensions of rings and modules satisfying some generalizations of injectivity or of related conditions. The study of these overrings, overmodules and extensions satisfying these conditions has been dealt with in detail in different papers. The question of when do certain properties transfer from any ring R to its many types of extensions such as matrix ring extensions, (skew-)group rings, polynomial ring extensions and Ore extensions, has also been of interest to many for a long time. It appears that the research work on various types of hulls and on the wide varieties of extensions is spread throughout the literature in disparate research papers. Thus, a book which presents (at least some part of) the state of the art on the subject and includes some of the most recent work done on these topics, is needed. That there has not been a comprehensive treatment of these topics, is one of the main reasons for us to write this research monograph.
Since the properties such hulls and extensions satisfy may be unlimited and thus cannot possibly be covered in one book, we wish to emphasize that in this monograph we have focused mainly on hulls and extensions that satisfy certain conditions on direct summands. We have however also made an attempt to provide a general theory for hulls belonging to arbitrary classes of rings (or modules) which can satisfy other properties.

Book Detail :-
Title: Extensions of Rings and Modules
Edition:
Author(s): Gary F Birkenmeier, Jae Keol Park, S Tariq Rizvi
Publisher: Springer-Verlag New York
Series:
Year: 2013
Pages: 442
Type: PDF
Language: English
ISBN: 978-0-387-92715-2,978-0-387-92716-9
Country: India
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About Author :-
Author Syed Tariq Rizvi is Professor and the Coordinator of Department of Mathematics, the Ohio State University (OSU), Lima, USA. After receiving his Ph.D. from McMaster University (Canada) in 1981 and teaching at the University of Waterloo (Canada), he joined OSU in 1982. A recipient of the Outstanding Scholar Award from OSU and a Fulbright Fellow Award from the Fulbright Board, Washington, DC, Professor Rizvi is world renowned for his research contributions to the theory of rings and modules. He has numerous research papers to his credit published in several international journals of high repute, published a monograph by Birkhäuser and coedited five research volumes with other reputed publishers. He also serves on editorial boards of ten international journals, including two as a managing or executive editor. Professor Rizvi has given over 120 invited talks at several national and international conferences, many as a plenary speaker. He has regularly organized sessions at a number of conferences including at the American Mathematical Society conferences and the Ohio State-Denison Math Conference series.


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Book Contents :-
Extensions of Rings and Modules written by Gary F Birkenmeier, Jae Keol Park, S Tariq Rizvi cover the following topics.
1. Preliminaries and Basic Results
2. Injectivity and Some of Its Generalizations
3. Baer, Rickart, and Quasi-Baer Rings
4. Baer, Quasi-Baer Modules, and Their Applications
5. Triangular Matrix Representations and Triangular Matrix Extensions
6. Matrix, Polynomial, and Group Ring Extensions
7. Essential Overring Extensions-Beyond the Maximal Ring of Quotients
8. Ring and Module Hulls
9. Hulls of Ring Extensions
10. Applications to Rings of Quotients and C*-Algebras
11. Open Problems and Questions
References
Index


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