Algebra: Chapter 8

This chapter discusses the study of modules and rings, including Noetherian and Artinian modules, radical, Morita equivalence, semisimple rings, Grothendieck groups, central simple algebras, and group algebras. It also provides a historical note on the evolution of these notions.

Format: Hardback
Length: 490 pages
Publication date: 16 March 2023
Publisher: Springer International Publishing AG

This book is a comprehensive English translation of an extensively revised edition of the eighth chapter of the book Algebra, the second Book of the Elements of Mathematics. It is dedicated to the study of specific classes of rings and modules, particularly focusing on the concepts of Noetherian or Artinian modules and rings, as well as that of radical. This chapter delves into the study of Morita equivalence of modules and algebras, providing an in-depth description of the structure of semisimple rings. Various Grothendieck groups are introduced, serving as universal tools for module invariants. The chapter also presents two particular cases of algebras over a field, showcasing the versatility of the theory. The theory of central simple algebras is explored in depth, involving the Brauer group, which is described in several detailed accounts. Finally, the chapter extends the general theory to representations of finite groups, demonstrating its broad applications. At the end of the volume, a historical note provides insights into the evolution of many of the developed notions throughout the book's history.

This book is a comprehensive English translation of an extensively revised edition of the eighth chapter of the book Algebra, the second Book of the Elements of Mathematics. It is dedicated to the study of specific classes of rings and modules, particularly focusing on the concepts of Noetherian or Artinian modules and rings, as well as that of radical. This chapter delves into the study of Morita equivalence of modules and algebras, providing an in-depth description of the structure of semisimple rings. Various Grothendieck groups are introduced, serving as universal tools for module invariants. The chapter also presents two particular cases of algebras over a field, showcasing the versatility of the theory. The theory of central simple algebras is explored in depth, involving the Brauer group, which is described in several detailed accounts. Finally, the chapter extends the general theory to representations of finite groups, demonstrating its broad applications. At the end of the volume, a historical note provides insights into the evolution of many of the developed notions throughout the book's history.

Weight: 922g
Dimension: 235 x 155 (mm)
ISBN-13: 9783031192920
Edition number: 1st ed. 2022