04729nam a22005775i 4500 978-3-031-50062-6 DE-He213 20251030111407.0 cr nn 008mamaa 240130s2024 sz | s |||| 0|eng d 9783031500626 978-3-031-50062-6 10.1007/978-3-031-50062-6 doi QA150-272 PBF bicssc MAT002000 bisacsh PBF thema Broué, Michel. author. aut http://id.loc.gov/vocabulary/relators/aut 70147 From Rings and Modules to Hopf Algebras [electronic resource] : One Flew Over the Algebraist's Nest / by Michel Broué. 1st ed. 2024. Cham : Springer International Publishing : Imprint: Springer, 2024. X, 533 p. 20 illus., 10 illus. in color. online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda 1 Prerequisites and Preliminaries -- Part I Rings and Modules -- 2 Rings, Polynomials, Divisibility -- 3 Polynomial Rings in Several Indeterminates -- 4 More on Modules -- 5 On Representations of Finite Groups -- Part II Integral Domains, Polynomials, Fields -- 6 Prime and Maximal Ideals, Integral Domains -- 7 Fields, Division Rings -- Part III Finitely Generated Modules -- 8 Integrality, Noetherianity -- 9 Finitely Generated Projective Modules -- 10 Finitely Generated Modules Over Dedekind Domains -- 11 Complement on Dedekind Domains -- Part IV Characteristic Zero Linear Representations of Finite Groups -- 12 Monoidal Categories: An Introduction -- 13 Characteristic 0 Representations -- 14 Playing With the Base Field -- 15 Induction and Restriction: Some Applications to Finite Groups -- 16 Brauer’s Theorem and Some Applications -- 17 Graded Representations and Characters -- 18 The Drinfeld–Lusztig Double of a Group Algebra. This textbook provides an introduction to fundamental concepts of algebra at upper undergraduate to graduate level, covering the theory of rings, fields and modules, as well as the representation theory of finite groups. Throughout the book, the exposition relies on universal constructions, making systematic use of quotients and category theory — whose language is introduced in the first chapter. The book is divided into four parts. Parts I and II cover foundations of rings and modules, field theory and generalities on finite group representations, insisting on rings of polynomials and their ideals. Part III culminates in the structure theory of finitely generated modules over Dedekind domains and its applications to abelian groups, linear maps, and foundations of algebraic number theory. Part IV is an extensive study of linear representations of finite groups over fields of characteristic zero, including graded representations and graded characters as well as a final chapter on the Drinfeld–Lusztig double of a group algebra, appearing for the first time in a textbook at this level. Based on over twenty years of teaching various aspects of algebra, mainly at the École Normale Supérieure (Paris) and at Peking University, the book reflects the audiences of the author's courses. In particular, foundations of abstract algebra, like linear algebra and elementary group theory, are assumed of the reader. Each of the of four parts can be used for a course — with a little ad hoc complement on the language of categories. Thanks to its rich choice of topics, the book can also serve students as a reference throughout their studies, from undergraduate to advanced graduate level. Algebra. 1501 Algebraic fields. 70148 Polynomials. 64269 Group theory. 70149 Algebra, Homological. 70150 Commutative algebra. 70151 Commutative rings. 70152 Algebra. 1501 Field Theory and Polynomials. 70153 Group Theory and Generalizations. 70154 Category Theory, Homological Algebra. 70155 Commutative Rings and Algebras. 70156 SpringerLink (Online service) Springer Nature eBook Printed edition: 9783031500619 Printed edition: 9783031500633 https://doi.org/10.1007/978-3-031-50062-6 ZDB-2-SMA ZDB-2-SXMS 57372 57372