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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
Contravariant hom functor      505
Contravariant power set functor      154
Contravariant representation theorem      149 515
Contravariant tensor      525
Contravariant tensors      525
Contravariant universal element      149 515
Contravariant vectors      525
Convergence      269
Convergent sequence      269
Converse relation      11 471
Convolution product      105
Coordinate function      208
Coordinate(s)      196 526 575
coordinates      573 596
Coordinates of a tensor      525
Coordinates of a vector      195
Coordinates of vectors      357
Copointed object      515
Coproduct      151 179
Coproduct object      151
Corepresentation      515
Correspondences, Galois      464
Coset      31 72ff
Counter image      76 154 476
Counterimage      76 154
Covariant      147 504
Covariant hom (functor)      133 192 506
Covariant hom functor      133 192 506
Covariant tensor      525ff
Covariant tensor (vector)      525ff
Cover in a relation      472 481
Cramer’s Rule      304
Critical point      350
Cubic polynomial      107
Cyclic group      51 53-55
Cyclic module      385
Cyclic permutation      63
Cyclic permutations      63
Cyclic primary decomposition theorem      394
Cyclic subgroup      53-55
Cyclic submodule      385
Cyclic subspace      389
Decomposable $p$-vector      552
Decomposable $p$-vectors      552
Decomposition      386
Decomposition into primes      116
Defining relation      61
Degree in graded modules      531
Degree of a field      442
Degree of a polynomial      107
Degree of algebraic element      440
Degree of modules, graded      531
Degree of polynomial      107
Delta, Kronecker      208 227 235 525
Dense subset      273
Denumerable      159
Dependence, linear      194
Derivative, formal      454
Derived subgroup      428
Descending central series      428
Determinant of an endomorphism      307 548
Determinant rank      305
Determinant(s)      293 297
Determinants rank      305
Diagonal argument      159
Diagonal form      345 404
Diagonal matrix      234 243 258 346
Diagram      16 146
Diamond isomorphism for modules      412
Diamond isomorphism theorem      411
Diamond Isomorphism Theorem for modules      412
Diamond isomorphism theorems      411 412
Difference      50 565
Difference of integers      26
Dihedral group      60
Dihedral groups      60
DIMENSION      199 200 241
Dimension in a poset      472
Dimension of annihilator      244
Dimension of kernel      202
Dimension posets      472
Direct product      62 179 409
Direct sum      92 179 183 204 243
Direct sum of matrices      243
Direct sum of matrix      243
Direct summand      183 204 382
Discriminant      268 438
Disjoint cycles      64
Disjoint sets      4
Disjoint union      152
Disjunctive canonical expression      451
Distance      286 354 586
Distributive      13
Distributive inequality      474
Distributive lattice      484
Distributive lattices      484
Distributive law      13
Distributive laws      85 160 165
Divisible      111
Division algebra      335
Division algorithm      29
Division ring      100 282
Domain      4
Domain of a function      4
Domain of a morphism      141 495
Domain of morphism      141 495
Domain with arrow      496
Double coset      75
Double dual      208
Double quotient isomorphism theorem      411
Dual base      357
Dual basis      208 233 357
Dual basis pair      235
Dual basis pairing      208 210
Dual linear transformation      188 357
Dual modules      185ff 357
Dual morphism of order      471
Dual of a diagram      147
Dual of a module      188
Dual of a morphism      185 186 188
Dual of biproduct      190
Dual of linear transformation      188 357
Dual of morphism      185 186 188
Dual of posets      471
Dual of right modules      186
Dual of tensor spaces      529
Dual of vector spaces      207 357
Dual order      145
Dual pairing      210
Dual(s)      207 500
Duality in categories      146
Duality in exterior algebra      555
Duality principle      471 500
Duals of biproducts      190
Duals of composite map      189
Duals of composite maps      189
Echelon      214
Eigenspace      258
Eigenvalue      257 311 365 376
Eigenvector      257 375
Eisenstein s criterion for irreducibility      124 451
Eisenstein’s irreducibility criterion      124 451
Element height      472
Element of a set      2
Element of modules, graded      531
Elementary column operation      401 402
Elementary column operations      401 402
Elementary divisors      391 394 395
Elementary divisors of matrix      391
Elementary equivalence      212 213
Elementary Jordan matrix      396
Elementary matrix      247 401
Elementary operations      212ff
Elementary row and column operations      401-402
Elementary row operations      402
Elementary symmetric function      460 468
Elementary symmetric functions      460 468
Empty set      2
Empty sets      2
Endomorphism      38 166 241
Endomorphism of algebra      334
Endomorphism ring      93
Endomorphism ring of an abelian group      93
Endomorphism vector spaces      407
Epic arrow      499
Epic morphism      499
Epimorphism      38 201
Equality of function      5
Equality of functions      1 5
Equality of sets      2
Equalizer      153
Equivalence class      33
Equivalence kernel      33
Equivalence relation      12 212
Equivalence under a group      70
Equivalent      413
Equivalent matrices      253 365 401ff
Equivalent object      499
Equivalent objects      499
Equivalent vectors      212 554
Euclidean      352 586 589 591
Euclidean algorithm      122-23 454
Euclidean equivalence      591
Euclidean group      589
Euclidean quadrics      591
Euclidean space(s)      586
Euclidean spaces      586
Euclidean vector space      352
Euclidean vector spaces      352
Euclid’s theorem      119
Evaluation      140 155
Evaluation of polynomial      108 109
Exact sequence      326 327 413 562
Exchange process      200
Expansion by cofactor      302
Expansion by cofactors      302
Exponential law      19
Exponents      52
Extension of a function      6
Extension of a group      409 413
Extension of a module      332
Extension of fields of finite degree      459
Extension of finite degree      433
Extension of function      6
Extension of subfield      442
Exterior algebra      543
Exterior product      543
Factor group      80
Field of quotient      101 130 265
Field of quotients      101 130 265
Field of rational function      122
Field of rational functions      122
Field of rational numbers      261
Field of scalars      193
Field of sets      485
Field(s)      99
Finite abelian group      62 387
Finite algebraic system      28
Finite combination      319
Finite dimensional space      200
Finite distributive lattice      485
Finite field      456
Finite group      142
Finite set      156 158
Finite sets      156 158
Finite span      168
Finite type      168 195 379
Finite-dimension space      200
Finite-dimensional vector spaces      207 241 252
Five lemma, short      327
Fixed field      462
Fixing subgroup      462
Forgetful functor      132 501
Form(s)      338ff
Formal derivative      454
Formal power series      284
Forms      255 584
Formula, binomial      90
Four group      46
Frame      573
Frattini argument      425
Free Boolean algebra      491ff
Free generator      196
Free module      175 196 224 296 496
Free modules      175 196 224 296 496
Free object      517
Free object functor      519
Function module      176
Function modules      176
Function ring      156
Function set      154
Function sets      154
Function space      205
Function(s)      1
Functional      573
functor      131 143 501ff
Functor categories      143 509 510
Fundamental theorem of algebra      280
Fundamental Theorem of Galois Theory      462
Galois correspondences      464
Galois field      457
Galois group      450 452 458
Gauss elimination      221
Gaussian integers      91
General associative law      41
General distributive law      87 586
General linear group      247
Generated ideal      538
Generation subfield      439
Generator of a group      58
Generator of an ideal      538
Generator of graded tensor algebra      540
Generators of an algebra      538
Graded algebra      533ff 534 539
Graded algebra of multilinear forms      536
Graded module      530 540
Graded modules      see Modules graded
Graded polynomial algebra      535-36
Graded quotient algebra      537
Graded subalgebra      535
Graded subalgebra, Boolean      535
Graded submodule      531
Graded tensor algebra      539ff
Graded tensor module      540
Gram - Schmidt orthogonalization      370
Graph      11
Grassmann algebra      535 536
Greater than      20
Greatest common divisor      434
Greatest common divisors      117
Greatest element      470
Greatest lower bound      145 271 473
Group extensions      409 413
Group halomorph      414
Group of automorphisms      466
Group of inner automorphisms      420
Group permutations      69
Group(s)      43 409ff 572 589 595
Hamilton      see Cayley - Hamilton Theorem
Height of element      472
1 2 3 4 5
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