Algebra: Chapter 8
Autor N. Bourbaki Traducere de Reinie Ernéen Limba Engleză Hardback – 16 mar 2023
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.
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Specificații
ISBN-13: 9783031192920
ISBN-10: 3031192923
Pagini: 490
Ilustrații: XVIII, 490 p.
Dimensiuni: 155 x 235 mm
Greutate: 0.89 kg
Ediția:2022
Editura: Springer International Publishing
Colecția Springer
Locul publicării:Cham, Switzerland
ISBN-10: 3031192923
Pagini: 490
Ilustrații: XVIII, 490 p.
Dimensiuni: 155 x 235 mm
Greutate: 0.89 kg
Ediția:2022
Editura: Springer International Publishing
Colecția Springer
Locul publicării:Cham, Switzerland
Cuprins
Artinian 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.
Textul de pe ultima copertă
This 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.
Caracteristici
Translation of a completely revised edition Introduces concepts absent in the previous editions Historical note at the end of the volume