## Algebra I: Chapters 1-3This 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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绝对值得翻译成中文，可惜没有。

### 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 |

669 | |

677 | |

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### Common terms and phrases

A-algebra A-linear A-module addition admits algebra apply associative automorphism basis belongs bijective called canonical canonical mapping commutative commutative ring consider consisting contained Conversely Corollary corresponding Deduce defined Definition denoted derived dimension direct sum distinct dual elements endomorphism equal equations equivalent example Exercise exists extension field finite follows formula Give given graded graduation hand hence homogeneous homomorphism ideal identified immediately implies injective integer inverse isomorphism Let G linear mapping matrix means module monoid morphism multiplication necessary and sufficient notation Note obtained particular permutation points projective properties Proposition prove quotient rational reduced relation Remark resp respect ring satisfying Show stable structure subgroup subgroup of G submodule subset subspace Suppose surjective taking tensor product Theorem Theory unique unit vector space whence write written zero