Classically Semisimple Rings
Springer International Publishing (Verlag)
978-3-031-14208-6 (ISBN)
The book presents the classical theory of semisimple rings from a modern, category-theoretic point of view. Examples from algebra are used to motivate the abstract language of category theory, which then provides a framework for the study of rings and modules, culminating in the Wedderburn-Artin classification of semisimple rings. In the last part of the book, readers are gently introduced to related topics such as tensor products, exchange modules and C*-algebras. As a final flourish, Rickart's theorem on group rings ties a number of these topics together. Each chapter ends with a selection of exercises of varying difficulty, and readers interested in the history of mathematics will find biographical sketches of important figures scattered throughout the text.
Assuming previous knowledge in linear and basic abstract algebra, this book can serve as a textbook for a course in algebra, providing students with valuable early exposure to category theory.
Martin Mathieu is Professor of Pure Mathematics at Queen's University Belfast. He received his Dr. rer.nat and his Habilitation from the University of Tubingen, where he was Privatdozent until 1998. After various spells at the University of the Saarland, Saarbrucken; the University of Iowa, Iowa City; and NUI Maynooth, Maynooth he moved to Northern Ireland where he has been living with his family since.
Introduction.- Chapter 1. Motivation from Ring Theory.- Chapter 2. Constructions with Modules.- Chapter 3. The Isomorphism Theorems.- Chapter 4. Noetherian Modules.- Chapter 5. Artinian Modules.- Chapter 6. Simple and Semisimple Modules.- Chapter 7. The Artin-Weddeburn Theorem.- Chapter 8. Tensor Products of Modules.- Chapter 9. Exchange Modules and Exchange Rings.- Chapter 10. Semiprimitivity of Group Rings.- Bibliography.- Index of Symbols.- Index.
"This book is an invaluable resource for graduate students and researchers in algebra and related fields, offering a detailed exposition of both classical results and modern developments in the theory of semisimple rings. The author's writing style is clear and concise ... . 'Classically Semisimple Rings' is an exceptional work and highly recommended to anyone interested in the theory of semisimple rings, making it a valuable resource for both students and researchers alike, and a testament to Mathieu's outstanding scholarship." (Jebrel M. Habeb, zbMATH 1519.16003, 2023)
"This is ... a well-written book, one that might find good use as a text for an elementary graduate course or a rather sophisticated undergraduate one. Faculty members may also find this to be a useful book to have on their shelves. I'm glad it's on mine." (Mark Hunacek, MAA Reviews, May 16, 2023)
| Erscheinungsdatum | 21.12.2022 |
|---|---|
| Zusatzinfo | XVI, 151 p. 37 illus. |
| Verlagsort | Cham |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Gewicht | 248 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Schlagworte | category theory in context • category theory textbook • graduate algebra textbook • introduction C*-algebras • introduction exchange rings • Jacobson radical • Martin Mathieu textbook • rings and modules textbook • semisimple rings book • Wedderburn-Artin theorem • Wedderburn–Artin theorem |
| ISBN-10 | 3-031-14208-X / 303114208X |
| ISBN-13 | 978-3-031-14208-6 / 9783031142086 |
| Zustand | Neuware |
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