Extensions of Rings and Modules

  • Gary F. Birkenmeier
  • Jae Keol Park
  • S Tariq Rizvi

Table of contents

  1. Front Matter
    Pages I-XX
  2. Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi
    Pages 1-17
  3. Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi
    Pages 19-60
  4. Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi
    Pages 61-92
  5. Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi
    Pages 93-137
  6. Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi
    Pages 139-187
  7. Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi
    Pages 189-215
  8. Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi
    Pages 217-266
  9. Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi
    Pages 267-326
  10. Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi
    Pages 327-354
  11. Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi
    Pages 355-407
  12. Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi
    Pages 409-411
  13. Back Matter
    Pages 413-432

About this book

Introduction

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.

Keywords

Algebra Baer C*-Algebras Functional Analysis Hull Matrix Theory Multilinear Algebra Triangular Matrix

Authors and affiliations

  • Gary F. Birkenmeier
    • 1
  • Jae Keol Park
    • 2
  • S Tariq Rizvi
    • 3
  1. 1., Dept. MathematicsUniversity of Louisiana, LafayetteLafayetteUSA
  2. 2.Dept. of MathematicsBusan National UniversityBusanKorea, Republic of (South Korea)
  3. 3., Dept. of MathematicsThe Ohio State UniversityLimaUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-0-387-92716-9
  • Copyright Information Springer Science+Business Media New York 2013
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-0-387-92715-2
  • Online ISBN 978-0-387-92716-9
  • About this book