LEADER 03574nam 2200433 450 001 996503549603316 005 20230516095950.0 010 $a9783031142093$b(electronic bk.) 010 $z9783031142086 035 $a(MiAaPQ)EBC7165640 035 $a(Au-PeEL)EBL7165640 035 $a(CKB)25913866300041 035 $a(PPN)267816634 035 $a(EXLCZ)9925913866300041 100 $a20230423d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aClassically semisimple rings $ea perspective through modules and categories /$fMartin Mathieu 210 1$aCham, Switzerland :$cSpringer,$d[2022] 210 4$d©2022 215 $a1 online resource (159 pages) 311 08$aPrint version: Mathieu, Martin Classically Semisimple Rings Cham : Springer International Publishing AG,c2022 9783031142086 320 $aIncludes bibliographical references and index. 327 $aIntro -- 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. 327 $a10.1 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. 606 $aAnells (Àlgebra)$2thub 608 $aLlibres electrònics$2thub 615 7$aAnells (Àlgebra) 676 $a512.4 700 $aMathieu$b Martin$01273316 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 912 $a996503549603316 996 $aClassically Semisimple Rings$93000277 997 $aUNISA