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Classically Semisimple Rings is a textbook on rings, modules and categories, aimed at advanced undergraduate and beginning graduate students.
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.
List of contents
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.
About the author
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 Tübingen, where he was Privatdozent until 1998. After various spells at the University of the Saarland, Saarbrücken; the University of Iowa, Iowa City; and NUI Maynooth, Maynooth he moved to Northern Ireland where he has been living with his family since.
Report
"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)
