# Algebra I: Chapters 1-3

Springer Science & Business Media, Aug 3, 1998 - Mathematics - 710 pages

This softcover reprint of the 1974 English translation of the first three chapters of Bourbaki’s Algebre gives a thorough exposition of the fundamentals of general, linear, and multilinear algebra. The first chapter introduces the basic objects, such as groups and rings. The second chapter studies the properties of modules and linear maps, and the third chapter discusses algebras, especially tensor algebras.

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### Contents

 To the Reader v 9 Theory of sets 9 Groups and groups with operators 30 Groups operating on a set 52 Extensions solvable groups nilpotent groups 65 Free monoids free groups 81 Rings 96 Commutative algebra 112
 Matrices 338 11 Graded modules and rings 363 Appendix Pseudomodules 378 Exercises for 2 386 Flat modules 2 Localization 3 Graduations nitrations and topo 388 Exercises for 3 395 Exercises for 5 398 413 Tensor Algebras Exterior Algebras Symmetric 427

 Fields 114 Exercises for 1 124 Exercises for 4 132 Exercises for 5 140 Exercises for 6 147 Exercises for 7 159 Exercises for 8 171 Exercises for 10 179 Algebra 191 Modules of linear mappings Duality 223 algebras symmetric algebras 4 Polynomials and rational fractions 232 Tensor products 243 modules 248 Relations between tensor products and homomorphism modules 267 Extension of the ring of scalars 277 Relations between restriction and extension of the ring of scalars 280 Extension of the ring of operators of a homomorphism module 282 Dual of a module obtained by extension of scalars 283 A criterion for finiteness 284 Direct limits of modules 286 Tensor product of direct limits 289 Vector spaces 292 Dimension of vector spaces 293 Dimension and codimension of a subspace of a vector space 295 Rank of a linear mapping 298 Dual of a vector space 299 Linear equations in vector spaces 304 Tensor product of vector spaces 306 Rank of an element of a tensor product 309 Extension of scalars for a vector space 310 Modules over integral domains 312 Restriction of the field of scalars in vector spaces 317 Rationality for a subspace 318 Rationality for a linear mapping 319 Rational linear forms 320 Application to linear systems 321 Smallest field of rationality 322 Criteria for rationality 323 Affine spaces and projective spaces 325 Barycentric calculus 326 Affine linear varieties 327 Affine linear mappings 329 Definition of projective spaces 331 Projective linear varieties 332 Projective completion of an affine space 333 Extension of rational functions 334
 Examples of algebras 438 Graded algebras 457 HomPE1 Fj gc HomcE2 471 factors 474 Tensor algebra Tensors 484 module Tensor algebra of a graded module 491 Symmetric algebras 497 Exterior algebras 507 Determinants 522 The AXmodule associated with an Amodule endo morphism 539 Characteristic polynomial of an endomorphism 540 Norms and traces 541 Properties of norms and traces relative to a module 542 General topology 543 Properties of norms and traces in an algebra 545 Discriminant of an algebra 549 Derivations 550 General definition of derivations 551 Functions of a real variable 553 Composition of derivations 554 Derivations of an algebra A into an Amodule 557 Derivations of an algebra 559 Functorial properties 560 Relations between derivations and algebra homomor phisms 561 Extension of derivations 562 noncommutative case 567 commutative case 568 Functorial properties of Kdifferentials 570 11 Cogebras products of multilinear forms inner products and duality 574 Coassociativity cocommutativity counit 579 Properties of graded cogebras of type N 584 Bigebras and skewbigebras 585 The graded duals TMr SM1 and A Mr 587 case of algebras 594 case of cogebras 597 case of bigebras 600 Inner products betweenTM andTMSM and SM 603 Explicit form of inner products in the case of a finitely generated free module 605 Isomorphisms between A M and A M for an n dimensional free module M 607 Application to the subspace associated with a vector 608 Pure vectors Grassmannians 609 Historical Note on Chapters II and III 655 Integration 658 Index of Notation 669 Index of Terminology 677 Copyright

### References to this book

 Several Complex Variables and the Geometry of Real HypersurfacesJohn P. D'AngeloLimited preview - 1993
 Topological Vector Spaces: Chapters 1-5N. BourbakiNo preview available - 2003
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### About the author (1998)

Nicolas Bourbaki is the pseudonym for a group of mathematicians that included Henri Cartan, Claude Chevalley, Jean Dieudonne, and Andres Weil. Mostly French, they emphasized an axiomatic and abstract treatment on all aspects of modern mathematics in Elements de mathematique. The first volume of Elements appeared in 1939. Subsequently, a wide variety of topics have been covered, including works on set theory, algebra, general topology, functions of a real variable, topological vector spaces, and integration. One of the goals of the Bourbaki series is to make the logical structure of mathematical concepts as transparent and intelligible as possible. The books listed below are typical of volumes written in the Bourbaki spirit and now available in English.