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Algebra,. Chapter 8

紀錄類型:	書目-語言資料,印刷品 : Monograph/item
正題名/作者:	Algebra,/ by N. Bourbaki.
作者:	Bourbaki, Nicolas.
出版者:	Cham :Springer International Publishing : : 2022.,
面頁冊數:	xviii, 490 p. :ill., digital ; : 24 cm.;
Contained By:	Springer Nature eBook
標題:	Algebra. -
電子資源:	https://doi.org/10.1007/978-3-031-19293-7
ISBN:	9783031192937

Algebra,. Chapter 8
Bourbaki, Nicolas.

Algebra,Chapter 8[electronic resource] /by N. Bourbaki. - Cham :Springer International Publishing :2022. - xviii, 490 p. :ill., digital ;24 cm.

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.

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.

ISBN: 9783031192937

Standard No.: 10.1007/978-3-031-19293-7doiSubjects--Topical Terms:

579870
Algebra.

LC Class. No.: QA152.3

Dewey Class. No.: 512

Algebra,. Chapter 8
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505 0 $a 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.
520 $a 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.
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