04520nam a22005295i 4500001001800000003000900018005001700027007001500044008004100059020003700100024003500137050001400172072001600186072002300202072001500225100008500240245012800325250001800453264007500471300006300546336002600609337002600635338003600661347002400697505094700721520170901668650001903377650002903396650002403425650002503449650003303474650003203507650003003539650001903569650004103588650004503629650004903674650004303723710003403766773002603800776003603826776003603862856004603898912001403944912001503958999001703973978-3-031-50062-6DE-He21320251030111407.0cr nn 008mamaa240130s2024    sz |    s    |||| 0|eng d  a97830315006269978-3-031-50062-67 a10.1007/978-3-031-50062-62doi 4aQA150-272 7aPBF2bicssc 7aMAT0020002bisacsh 7aPBF2thema1 aBroué, Michel.eauthor.4aut4http://id.loc.gov/vocabulary/relators/aut97014710aFrom Rings and Modules to Hopf Algebrash[electronic resource] :bOne Flew Over the Algebraist's Nest /cby Michel Broué.  a1st ed. 2024. 1aCham :bSpringer International Publishing :bImprint: Springer,c2024.  aX, 533 p. 20 illus., 10 illus. in color.bonline resource.  atextbtxt2rdacontent  acomputerbc2rdamedia  aonline resourcebcr2rdacarrier  atext filebPDF2rda0 a1 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.  aThis 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. 0aAlgebra.91501 0aAlgebraic fields.970148 0aPolynomials.964269 0aGroup theory.970149 0aAlgebra, Homological.970150 0aCommutative algebra.970151 0aCommutative rings.97015214aAlgebra.9150124aField Theory and Polynomials.97015324aGroup Theory and Generalizations.97015424aCategory Theory, Homological Algebra.97015524aCommutative Rings and Algebras.9701562 aSpringerLink (Online service)0 tSpringer Nature eBook08iPrinted edition:z978303150061908iPrinted edition:z978303150063340uhttps://doi.org/10.1007/978-3-031-50062-6  aZDB-2-SMA  aZDB-2-SXMS  c57372d57372