## Algebra I: Chapters 1-3This is the softcover reprint of the English translation of 1974 (available from Springer since 1989) of the first 3 chapters of Bourbaki's 'Algebre'. It gives a thorough exposition of the fundamentals of general, linear and multilinear algebra. The first chapter introduces the basic objects; groups, actions, rings, fields. The second chapter studies the properties of modules and linear maps, especially with respect to the tensor product and duality constructions. The third chapter investigates algebras, in particular tensor algebras. Determinants, norms, traces and derivations are also studied. |

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绝对值得翻译成中文，可惜没有。

### Contents

Description of formal mathematics 2 Theory of sets 3 Ordered sets | 1 |

Groups and groups with operators | 30 |

Groups operating on a set | 52 |

Extensions solvable groups nilpotent groups | 65 |

Associated prime ideals and primary decomposition 5 Integers | 81 |

Rings | 96 |

Fields | 114 |

Exercises for 1 | 124 |

Matrices | 338 |

11 Graded modules and rings | 363 |

Appendix Pseudomodules | 378 |

Exercises for 2 | 386 |

Exercises for 3 | 395 |

Exercises for 5 398 | 413 |

Tensor Algebras Exterior Algebras Symmetric | 427 |

Examples of algebras | 438 |

Exercises for 4 | 132 |

Exercises for 5 | 140 |

Exercises for 6 | 147 |

Exercises for 7 | 159 |

Exercises for 8 | 171 |

pact spaces 4 Extension of a measure Lp spaces 5 Integration of mea | 177 |

Exercises for 10 | 179 |

Linear Algebra | 191 |

Modules of linear mappings Duality | 227 |

Tensor products | 243 |

Relations between tensor products and homomorphism modules | 267 |

Trace of an endomorphism | 273 |

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 |

Graded algebras | 457 |

The homomorphism | 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 |

Norm and trace in an algebra | 543 |

Properties of norms and traces in an algebra | 545 |

Discriminant of an algebra | 549 |

Derivations | 550 |

General definition of derivations | 551 |

Differential and Analytic Manifolds | 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 |

669 | |

677 | |

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### Common terms and phrases

A-algebra A-linear A-module structure algebra associative basis bijective called canonical homomorphism canonical injection canonical mapping commutative group commutative ring compatible Deduce defined Definition denoted direct sum direct system endomorphism equivalence relation exact sequence Exercise exists finite group formula graded group G group with operators hence Hom(E HomA(E homogeneous identified identity element identity mapping implies indexing set induction integer inverse isomorphism kernel law of composition left A-module left ideal left resp Lemma Let G Let H linear mapping magma matrix module monoid morphism multiplication necessary and sufficient nilpotent non-empty non-zero normal stable subgroup normal subgroup notation ordered pair permutation prime number quotient group respect right A-module ring homomorphism Set Theory Show stable subgroup stable subset subgroup of G submodule subring subspace Suppose surjective Sylow subgroup tensor product Theorem two-sided ideal unique vector space whence