Classically semisimple rings : a perspective through modules and categories / / Martin Mathieu

• Record Nr.: UNISA996503549603316
• Autore: Mathieu Martin
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2022]; ©2022
• ISBN: 9783031142093; 9783031142086
• Descrizione fisica: 1 online resource (159 pages)
• Disciplina: 512.4
• Soggetti: Anells (Àlgebra); Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- 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.