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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
Natural isomorphism      507
Natural map      209
Natural numbers      15
Natural transformation      507
Negative      50
Negative definite      349
Nilpotent group      427
Nilpotent linear transformation      313
Nilpotent matrix      313
Noetherian      378ff
Noetherian module      379
Noetherian modules      378ff
Noetherian ring      379 380
Nonarchimedean      285
Nondegenerate critical point      350
Nondegenerate form      347
Nonmodular lattice      480
Nonmodular lattices      480
Nonsingular matrix      246
Norm of a form      358
Norm of a vector      353
Normal extension      458ff
Normal linear transformation      374
Normal matrix      377ff
Normal subgroup      77 424
Normalized vector      356
Null set      2
Null sets      2
Null space      203
Nullary operations      13 489 497
Nullity of a map      203
Nullity of linear map      203
Nullity of matrix      244
Object function      501
Object(s)      64
Objects of categories      141
Opposite binary operation      13
Opposite categories      505
Opposite category      505
Opposite group      49
Opposite ring      185
Opposite rings      185
Orbit      64 70
Orbits of group      70
Order      51 284
Order automorphism      275
Order automorphisms      275
Order isomorphism      264 266
Order morphism of rings      262
Order of a module      381-93
Order of an element in a group      53
Order of elements in group      51
Order-morphism of rings      262
Ordered domain      261 295
Ordered field      261 265ff 285
Ordered pair      10
Ordered ring      261ff
Ordered rings      261ff
Ordered set      22
Ordered sets      22 471
Ordered triple      10
Oriented vector space      308
Ornaments      61
Orthogonal column      361
Orthogonal complement      353 370 375
Orthogonal equivalence      363
Orthogonal Gram - Schmidt process      370
Orthogonal group      360 362
Orthogonal linear transformation      360
Orthogonal map      361
Orthogonal matrix      361
Orthogonal vectors      353 370
Orthogonally equivalent matrix      365
Orthonormal      355 356 589
Orthonormal base      357
Orthonormal basis      356 370
Orthonormal column      361
Orthonormal frame      589
Orthonormal list      355 376
Outer automorphism      83
Outer automorphisms      83
Parameter Theorem      514
Parity of a permutation      66 295
Parity of permutations      66 295
Partial order      143ff 471
Partially ordered sets      471
Partition      3
Peano axiom      15
Peano postulates      15
Permutation group      69
Permutation matrix      247
permutations      44 295 468
Permutations matrix      247
Plane      276 563 592 594
Point of a poset      472
Point(s)      561 565 592ff
Pointed object      513
Pointed objects      513
Pointed ring      138 514
Pointed rings      138 514
Points, difference of      565
Points, of affine line      561
Points, of projective spaces      592ff
Pointwise addition      166
Pointwise addition of morphisms      166
Pointwise product      92
Pointwise sum      105 205
polarities      464
Polarize      580
Polarize a quadratic form      343 370
Polarized form      344
Polynomial algebra      536
Polynomial function      109 269 338
Polynomial indeterminates      139
Polynomial ring      109 130
Polynomial(s)      104ff
Poset of subset      464
Posets      144 470ff
Posetsof subsets      464
Positive      349 369
Positive definite      349
Positive elements      261
Positive integers      28
Positive orientation      308
Positive semi-definite      349
Positive semidefinite      349
Power set      153
Power sets      153
Power(s)      86
Primary decomposition theorem      393
Primary decomposition theorems      393 394
Primary modules      392
Prime divisors      111
Prime field      120
Prime ideal      125
Prime subfield      121 266
Principal axis theorem      365
Principal ideal      96 115-16 384 389 392 405
Principal ideal domain      115-16 384 389 392
Principle of Mathematical Induction      15
PRODUCT      92 179
Product categories      503
Product category      503
Product morphism      150
Product morphisms      150
Product object      150
Product of elements      534
Product of graded modules      531
Product of group      43 46
Product of groups      43 46
Product of linear forms      339
Product of natural numbers      18
Product of polynomials      105
Product of posets      478
Product of quaternion numbers      282
Product of rings      92
Product of sets      10
Product of transpositions      296
Projection of biproducts      179
Projection of graded modules      530 540
Projection of products      150
Projection of quotient      30 80 531
Projection of quotient groups      80
Projection of quotient sets      33
Projective      592 594ff
Projective conic      597
Projective group      595 596
Projective isomorphism      595
Projective line      594
Projective plane      592 594
Projective quadric      597
Projective quadrics      597
Projective space      594ff
Projective special linear group      435
Projective subspace      594
Projective subspaces      594
Projective transpose      481
Projectivespace(s)      594ff
Proper subgroup      56
Proper submodule      168
Proper subset      2
Proper value      258
Proper vector      258
Proper vectors      258
Pseudopolynomial, $p$-adic      287
Quadratic field      187
Quadratic form      338 343ff 580
Quadratic functional      580
Quadratic polynomial      107 345
Quadric      584 590 591 597
Quadric hypersurfaces      591
Quadric, euclidean      590
Quadrilinear function      524
Quartic polynomial      107
Quaternion      281ff
Quaternion group      62 426
Quintic polynomial      107
Quotient algebra      537
Quotient group      79ff 130 409ff 435 463
Quotient module      172 481 531
Quotient modules      172 481 538
Quotient of biproducts      183
Quotient ring      97 130 277
Quotient set      33 130
Quotient sets      33 130
Quotient(s)      133
R-modules      160
Radical      582
Radical extension      467
Radicals, solution of equations by      437 465ff
Radius of convergence      284
RANGE      203
Rank      582
Rank of a map      203
Rank of bilinear form      340
Rank of free modules      306 398 556
Rank of linear map      203
Rank of matrix      244
Rank of morphism      401
Rank of quadratic form      343
Rank of quadratic functional      582
Rational canonical form      391
Rational field      261 271ff
Rational functions      122
Rational numbers      261ff 272 273
Rational polynomial      280
Rational relations      451
Rational-form      391
Real axis      277
Real complete field      271
Real field      261 271ff
Real numbers      272 273
Real polynomial      280
Real simple algebra      335
Real symmetric matrix      349
Rectangular matrix      226
Recurrent $p$-adic series      291
Recurrent coefficients      288
Reduced echelon      218
Reduced echelon form      218
Refinement of a series      430
Reflection      249
Reflexive relation      12 470
Reflexivity      470
Regular representation      69 72
Relative complement      484
Relatively prime      117 383 392 466
Relatively prime factorization      392
Relatively prime order      117 383 466
Representation      69 512
Representation of a functor      512
Representation of a group      503
Representation theorem for universal elements      137 330
Representing object      513
Residue class integer      31
Residue class ring      97
Restriction of a function      6
Restriction of function      6
Right $R$-module      185
Right adjoint      518
Right cancellation      48
Right coset      74
Right inverse      7 48
Right inverse linearity      191
Right multiplication      72
Right regular representation      72
Right scalar      231
Right unit      48
Rigid map      588
Rigid motion      588
Ring automorphisms      275
Ring of integers      94
Ring of polynomials      109 130
Ring of quaternion      282
Ring of quaternions      282
Ring of scalars      160
Ring of sets      485
Ring(s)      85ff
Root of polynomial      150
Roots of equations      465
Roots of polynomial equation      113
Roots of polynomials      150
Roots of unity      279
Rotation      230
Row      187
Row matrix      228
Row operations      221 402
Row rank      245
Saddle point      350
scalar      160 187
Scalar coefficients      301
Scalar factors      339
Scalar matrix      205 228 243 339
Scalar multiple      160 298 332
Scalar multiplication      160
Scalar product      351
Scalars      160
Schwartz inequality      353 370
Schwarz inequality      353 370
Second principle of induction      22
1 2 3 4 5
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