Algebra I: Chapters 1-3

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Springer Science & Business Media, Aug 3, 1998 - Mathematics - 710 pages
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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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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
Index of Notation
669
Index of Terminology
677
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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.