LEADER 03302nam 2200505 450 001 9910682564303321 005 20240223125115.0 010 $a3-031-19293-1 024 7 $a10.1007/978-3-031-19293-7 035 $a(MiAaPQ)EBC7216418 035 $a(Au-PeEL)EBL7216418 035 $a(CKB)26270927000041 035 $a(DE-He213)978-3-031-19293-7 035 $a(PPN)269098461 035 $a(EXLCZ)9926270927000041 100 $a20230729d2023 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aAlgebra $eChapter 8 /$fN. Bourbaki and Reinie Erne? 205 $a1st ed. 2022. 210 1$aCham, Switzerland :$cSpringer Nature Switzerland AG,$d[2023] 210 4$d©2023 215 $a1 online resource (505 pages) 311 08$aPrint version: Bourbaki, N. Algebra Cham : Springer International Publishing AG,c2023 9783031192920 320 $aIncludes bibliographical references. 327 $aArtinian Modules and Noetherian Modules -- The Structure of Modules of Finite Length -- Simple Modules -- Semisimple Modules -- Commutation -- Morita Equivalence of Modules and Algebras -- Simple Rings -- Semisimple Rings -- Radical -- Modules over an Artinian Ring -- Grothendieck Groups -- Tensor Products of Semisimple Modules -- Absolutely Semisimple Algebras -- Central Simple Algebras -- Brauer Groups -- Other Descriptions of the Brauer Group -- Reduced Norms and Traces -- Simple Algebras over a Finite Field -- Quaternion Algebras -- Linear Representations of Algebras -- Linear Representations of Finite Groups -- Algebras without Unit Element -- Determinants over a Noncommunitative Field -- Hilbert's Nullstellensatz -- Trace of an Endomorphism of Finite Rank -- Historical Note -- Bibliography -- Notation Index -- Terminology Index. 330 $aThis book is an English translation of an entirely revised version of the 1958 edition of the eighth chapter of the book Algebra, the second Book of the Elements of Mathematics. It is devoted to the study of certain classes of rings and of modules, in particular to the notions of Noetherian or Artinian modules and rings, as well as that of radical. This chapter studies Morita equivalence of module and algebras, it describes the structure of semisimple rings. Various Grothendieck groups are defined that play a universal role for module invariants. The chapter also presents two particular cases of algebras over a field. The theory of central simple algebras is discussed in detail; their classification involves the Brauer group, of which several descriptions are given. Finally, the chapter considers group algebras and applies the general theory to representations of finite groups. At the end of the volume, a historical note taken from the previous edition recounts the evolution of many of the developed notions. 606 $aAlgebra 606 $aÀlgebra$2thub 608 $aLlibres electrònics$2thub 615 0$aAlgebra. 615 7$aÀlgebra 676 $a512.4 700 $aBourbaki$b N.$00 702 $aErne?$b Reinie 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910682564303321 996 $aAlgebra$93406430 997 $aUNINA