Classically Semisimple Rings: A Perspective Through Modules and Categories

Description

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. Preface Contents Introduction 1 Motivation from Ring Theory 1.1 Basics on Modules 1.1.1 Reducing Complicated Rings to Simpler Ones 1.1.2 One-Sided Ideals in Noncommutative Rings 1.1.3 Images of Ideals Under Homomorphisms 1.1.4 Embedding into the Endomorphism Ring and General Representations 1.1.5 Group Representations 1.2 Categories of Modules 1.3 Exercises 2 Constructions with Modules 2.1 Some Special Morphisms 2.2 Quotient Modules 2.3 Generating Modules 2.4 Direct Sums and Products of Modules 2.5 Free Modules 2.6 Special Objects in a Category 2.6.1 Free Objects 2.6.2 Products and Coproducts 2.7 Exercises 3 The Isomorphism Theorems 3.1 Isomorphisms Between Modules 3.2 Functors and Natural Transformations 3.3 Exercises 4 Noetherian Modules 4.1 Permanence Properties of Noetherian Modules 4.2 Exact Categories and Exact Functors 4.2.1 Kernels and Cokernels 4.2.2 Exact Categories 4.2.3 Exact Functors 4.3 Exercises 5 Artinian Modules 5.1 Finitely Cogenerated Modules 5.2 Commutative Artinian Rings 5.3 Artinian vs. Noetherian Modules 5.4 Abelian Categories 5.5 Exercises 6 Simple and Semisimple Modules 6.1 Decomposition of Modules 6.2 Projective and Injective Modules 6.3 Projective and Injective Objects 6.4 Exercises 7 The Artin–Wedderburn Theorem 7.1 The Structure of Semisimple Rings 7.2 Maschke's Theorem 7.3 The Hopkins–Levitzki Theorem 7.4 Exercises 8 Tensor Products of Modules 8.1 Tensor Product of Modules 8.2 Tensor Product of Algebras 8.3 Adjoint Functors 8.4 Exercises 9 Exchange Modules and Exchange Rings 9.1 Basic Properties of Exchange Modules 9.2 Exchange Rings 9.3 Commutative Exchange Rings 9.4 Exercises 10 Semiprimitivity of Group Rings 10.1 Basic Properties 10.2 Some Analytic Structure on ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (double struck upper C left bracket upper G right bracket) /StPNE pdfmark [/StBMC pdfmarkC[G]ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark 10.3 The Semiprimitivity Problem 10.4 Exercises Bibliography Index of Symbols Index