Extensions of Rings and Modules

The focus of this monograph is the study of rings and modules which have a rich supply of direct summands with respect to various extensions. The first four chapters of the book discuss rings and modules which generalize injectivity (e.g., extending modules), or for which certain annihilators become direct summands (e.g., Baer rings). Ring extensions such as matrix, polynomial, group ring, and essential extensions of rings from the aforementioned classes are considered in the next three chapters. A theory of ring and module hulls relative to a specific class of rings or modules is introduced and developed in the following two chapters. While applications of the results presented can be found throughout the book, the final chapter mainly consists of applications to algebra and functional analysis. These include obtaining characterizations of rings of quotients as direct products of prime rings and descriptions of certain C*-algebras via (quasi-)Baer rings.

Extensions of Rings and Modules introduces for the first time in book form:

* Baer, quasi-Baer, and Rickart modules
* The theory of generalized triangular matrix rings via sets of triangulating idempotents
* A discussion of essential overrings that are not rings of quotients of a base ring and Osofsky's study on the self-injectivity of the injective hull of a ring
* Applications of the theory of quasi-Baer rings to C*-algebras

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.

List of contents

Preliminaries and Basic Results.- Injectivity and Some of Its Generalizations.- Baer, Rickart, and Quasi-Baer Rings.- Baer, Quasi-Baer Modules, and Their Applications.- Triangular Matrix Representations and Triangular Matrix Extensions.- Matrix, Polynomial, and Group Ring Extensions.- Essential Overring Extensions - Beyond the Maximal Ring of Quotients.- Ring and Module Hulls.- Hulls of Ring Extensions.- Applications to Rings of Quotients and C* Algebras.- Open Problems and Questions.- References.- Index.

Summary

The focus of this monograph is the study of rings and modules which have a rich supply of direct summands with respect to various extensions. The first four chapters of the book discuss rings and modules which generalize injectivity (e.g., extending modules), or for which certain annihilators become direct summands (e.g., Baer rings). Ring extensions such as matrix, polynomial, group ring, and essential extensions of rings from the aforementioned classes are considered in the next three chapters. A theory of ring and module hulls relative to a specific class of rings or modules is introduced and developed in the following two chapters. While applications of the results presented can be found throughout the book, the final chapter mainly consists of applications to algebra and functional analysis. These include obtaining characterizations of rings of quotients as direct products of prime rings and descriptions of certain C*-algebras via (quasi-)Baer rings.
 
Extensions of Rings and Modules introduces for the first time in book form:

* Baer, quasi-Baer, and Rickart modules  
* The theory of generalized triangular matrix rings via sets of triangulating idempotents
* A discussion of essential overrings that are not rings of quotients of a base ring and Osofsky's study on the self-injectivity of the injective hull of a ring
* Applications of the theory of quasi-Baer rings to C*-algebras
 
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.

Additional text

“The authors of this monograph have made important contributions to this subject during the last few decades and they aim to give a comprehensive treatment of the topic. … Each chapter is followed by brief historical notes. There is a large number of exercises and a bibliography with over 400 entries. This volume fills a gap in the literature and will be an important reference for ring theorists in years to come.” (C. Baxa, Monatshefte für Mathematik, Vol. 180, 2016)
“This book is to present the state of the art on the subject, which includes some of the most recent work done on these topics. … The monograph is as self-contained as possible, with extensive references that are useful for further study. The book is very well written and it can serve as a good tool for graduate students, ting theorists and researchers in algebra and analysis.” (J. Jirásko, Mathematical Reviews, September, 2014)
“This book is a research monograph intended for research mathematicians and advanced graduate students in ring theory. … the book is well-organized, well-written and informative. It will be a handy companion for ring theorists in general, but an essential tool for researchers who want to extend our knowledge of extensions, hulls and the transfer of properties between rings and modules and their extensions.” (Stefan Veldsman, zbMATH, Vol. 1291, 2014)

Report

"The authors of this monograph have made important contributions to this subject during the last few decades and they aim to give a comprehensive treatment of the topic. ... Each chapter is followed by brief historical notes. There is a large number of exercises and a bibliography with over 400 entries. This volume fills a gap in the literature and will be an important reference for ring theorists in years to come." (C. Baxa, Monatshefte für Mathematik, Vol. 180, 2016)

"This book is to present the state of the art on the subject, which includes some of the most recent work done on these topics. ... The monograph is as self-contained as possible, with extensive references that are useful for further study. The book is very well written and it can serve as a good tool for graduate students, ting theorists and researchers in algebra and analysis." (J. Jirásko, Mathematical Reviews, September, 2014)
"This book is a research monograph intended for research mathematicians and advanced graduate students in ring theory. ... the book is well-organized, well-written and informative. It will be a handy companion for ring theorists in general, but an essential tool for researchers who want to extend our knowledge of extensions, hulls and the transfer of properties between rings and modules and their extensions." (Stefan Veldsman, zbMATH, Vol. 1291, 2014)

Product details

• Authors: Gary Birkenmeier, Gary F Birkenmeier, Gary F. Birkenmeier, Jae Keo Park, Jae Keol Park, S Tariq Rizvi, SYED T RIZVI, Syed Tariq Rizvi
• Publisher: Springer, Berlin
• Languages: English
• Product format: Hardback
• Released: 25.07.2013
• EAN: 9780387927152
• ISBN: 978-0-387-92715-2
• No. of pages: 432
• Dimensions: 163 mm x 241 mm x 30 mm
• Weight: 836 g
• Illustrations: XX, 432 p.
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Arithmetic, algebra; Algebra, B, Mathematics and Statistics, Functional Analysis, Linear Algebra, Matrix theory, Linear and Multilinear Algebras, Matrix Theory, Functional analysis & transforms