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Mac Lane S., Birkhoff G.D. — Algebra
Mac Lane S., Birkhoff G.D. — Algebra

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Название: Algebra

Авторы: Mac Lane S., Birkhoff G.D.

Аннотация:

This book presents modern algebra from first principles and is accessible to undergraduates or graduates. It combines standard materials and necessary algebraic manipulations with general concepts that clarify meaning and importance.
This conceptual approach to algebra starts with a description of algebraic structures by means of axioms chosen to suit the examples, for instance, axioms for groups, rings, fields, lattices, and vector spaces. This axiomatic approach — emphasized by Hilbert and developed in Germany by Noether, Artin, Van der Waerden, et al., in the 1920s — was popularized for the graduate level in the 1940s and 1950s to some degree by the authors' publication of A Survey of Modern Algebra. The present book presents the developments from that time to the first printing of this book. This third edition includes corrections made by the authors.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 3 edition

Год издания: 1999

Количество страниц: 626

Добавлена в каталог: 10.04.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$K$-algebra      334ff
$K$-linear transformation tensor product      329
$K’$ modules extensions      332
$n$-ary operation      13
$n$-ary operations      13
$p$-adic integers      280 287
$p$-adic numbers      286ff
$p$-adic order      287
$p$-adic series      287ff
$p$-group      421
$p$-module      392
$p$-vector      552
$R$-modules      160 185
$Z$-modules      162
$\mathcal{K}$-copointed object      515
Abelian group      50 160 162 387 416
Absolute value      264 278
Absorption law      473
Action by conjugation      70
Action of a group      70 502 564
Addition      18
Addition of natural numbers      18 24
Additive abelian group      160
Additive categories      228
Additive group      26 40 43 94 165
Additive group of morphisms      165
Additive inverse      26
Additive monoid      40
Additive monoids      40
Additive morphism      163
Additive unit      50
Adjoint      358 372
Adjoint functor      495 517ff 545
Adjoint matrix      372
Adjoint of map      372
Adjoint space      369
Adjoint transformation      358
Adjunction      517
Affine automorphism      561
Affine axioms      564 569
Affine equivalence      584
Affine functional      573 580
Affine group      561 572
Affine independence      575
Affine line      561
Affine plane      563
Affine property      562
Affine quadrics      584
Affine space      564 598
Affine space(s)      564 598
Affine subspace      576
Affine subspaces      576
Algebra of multilinear forms      536
Algebra(s)      334 532ff
Algebraic element      439
Algebraic extension      456
Algebraic number      440 444
Algebraic system      39 334 495
Algebraic systems      28 39 334 495
Algebraically independent      442
Alias-alibi      582 584
Alternating      546
Alternating bilinear form      341
Alternating form      341 396 558
Alternating form(s)      341 396 558
Alternating group      67 409 468
Alternating matrix      341
Alternating multilinear function      295 546
Angle between vectors      351 354
Annihilator      209
Annihilator of subspaces      210
Antiautomorphism      230
Antisymmetric relation      144 470
Antisymmetry      144 470
Archimedean Law      272
Argument of a complex number      278
Argument of function      4
Arithmetic, $p$-adic      290
arrow      2 4
Arrow categories      498
Arrow category      496 498
Artin lemma      460
Ascending central series      427
Ascending chain condition      152 379
Ascending chain condition on submodule      379
Ascending sequence      380
ASSIGNMENT      5
Associate      111
Associative      13
Associative composition      496
Associative law      6 13 473
Atoms of a poset      472
Automorphism classes      82
Automorphism ring      275
Automorphism(s)      38 212 247 561
Average      562 566 576
Axiom for a group      47
Axiom for a module      160
Axiom of Choice      8
Axiom(s)      61
Barred arrow      5
Barycentric coordinates      575
Base field      440
Base for vector spaces      198
Base(s)      115 241
Basis      194 197
Basis for free modules      175
Basis of ideal      115
Basis of ideals      115
Betti number      399
Biadditive      294
Biadditive function      324
Biaffine functional      578
Bifunctor      155 503
Bijection      6 225 410 513 561
Bijection function      6
Bilinear form      338ff
Bilinear form(s)      338ff
Bilinear function      293 318-19 324 522
Bilinear map      319 557
Bimodule      186
Binaiy relation      11 471
Binary operation      12 13 37 289 473
Binary operations      12 13 37 289 473
Binary relation      11 471
Binomial coefficients      90
Binomial formula      90
Biproduct      152 183 232
Biproduct of modules      179 386
Block matrix      243
Block multiplication      242
Boolean      487ff
Boolean algebra      476 488
Boolean lattices      476 488
Boolean polynomial      491
Boolean subalgebra      489
Bounded set      276
Bounded sets      276
Burnside’s Theorem      422
Cancellation      18 48 99
Cancellation law      48 99
Canonical disjunctive-expression      451
Canonical form(s)      255
Canonical isomorphism      523
Cardinal numbers      156 158
Cartesian product      10 179
Cartesian product diagram      10
Cartesian product functor      519
Cartesian product of function      10
Cartesian product of functions      10
Cartesian product of sets      10
Categories      133 141ff 495ff 570
Categories of all categories      501
Categories of groups      142 497
Categories of modules      163 497
Categories of monoids      142 497
Categories of pointed sets      497
Categories of rings      142 497
Categories of sets      141 153 497
Categories of vector spaces      234
Category of affine spaces      570
Category of algebra      334
Category of finite groups      142
Category of group      142 497
Category of modules      163 497
Category of monoids      142 497
Category of ring      142 497
Category of sets      141 153 497
Cauchy sequence      270 273-74 286-87 288
Cauchy theorem      421
Cayley - Hamilton theorem      315 391
Cayley theorem      69
Center of a group      418
Center of a subgroup      427
Center of quadratic functional      582
Central conic      350 366
centralizer      428
Chain      470 484
Change matrix      251ff 340
Change of base      25 223 251ff 340
Change of bases      251
Change of coordinates      251
Change of ring      331
Change of rings      331
Character group      416
Character groups of abelian group      416
Characteristic function      154 507
Characteristic of field      455
Characteristic of integral domain      120
Characteristic of integral domains      120
Characteristic of polynomial      309ff
Characteristic of ring      88
Characteristic of rings      88
Characteristic polynomial      309ff
Characteristic polynomial of matrix      311
Characteristic subgroup      411 417ff
Characteristic value      258
Characteristic vector      258
Chinese remainder theorem      125
Class equation      421
Class equations      421
Class of nilpotency      427
Class of objects      496
Classical adjoint      303
Closed      56
Closed below      485
Closed below subset      485
Closed set      477
Closed sets      477
Closure      56 57 90 576
Closure property      477
Closure under multiplication      56
Codomain      4 81 141 495
Codomain of an arrow      496
Codomain of function      4
Codomain of morphism      141 495
Coefficients of polynomial      107
Cofactor      301
Cokernel      410
column      187 242 361
Column and row equivalence      213ff
Column equivalence      213
Column matrix      224 228
Column operations      213
Column rank      245
Combination, linear      194 234 552
Commutative      13 473
Commutative algebra      335
Commutative diagram      16 147 238
Commutative graded algebra      536
Commutative group      50
Commutative law      13 473
Commutative monoid      40 41
Commutative monoids      40 41
Commutative ring      85 139 162
Commutativity of triangular diagram      5
Commutator      83 418 428
Commutator subgroup      418 428
Companion matrix      314 390
Complement      3 82 550
Complement in a lattice      487
Complement in lattices      487
Complete invariant      254
Complete invariants      254
Complete lattice      477
Complete lattices      477
Complete metric space      287
Complete ordered field      272
Complete systems of invariants      254
Complex conjugate      271 368
Complex field      277
Complex number      397
complex numbers      397
Complex plane      277
Components of a tensor      525
Composite arrow      496
Composite function      5
Composite functor      518
Composite list      201
Composite matrix      549
Composite matrix of a composite      240ff
Composite morphism      226 232
Composite relation      11
Composite transformation      509
Composition factors      432
Composition of transformation      69
Composition series      430
Concentrated in degree zero      532
Concrete categories      141 227 334 495 497
Concrete category      141 227 334 495 497
Congruence of matrices      342 366
Congruent matrix      342 366
conic      597
Conic, central      350 366
conjugate      70
Conjugate class      71 420 421
Conjugate equations      421
Conjugate equivalence      33
Conjugate linear form      368
Conjugate module      186
Conjugate modules      186
Conjugate symmetry      369 376
Conjugate transpose      372
Conjugate transpose matrix      372
Conjugation      65 70 419 450
Connected chain      481
Consistency      473
Consistency condition      473
Constant coefficients      187
Constant function      92-93 110
Constant functional      578
Constant polynomial      107 108
Constant sequence      287
Construction by ruler and compass      445-46
Continuous polynomial function      269
Contraction      528
Contraction of a tensor      527
Contravariant      147
Contravariant functor      147 155 504ff
Contravariant hom (functor)      505
1 2 3 4 5
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