Algebra

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.