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Abhyankar S.S. — Lectures on Algebra Volume 1
Abhyankar S.S. — Lectures on Algebra Volume 1



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Название: Lectures on Algebra Volume 1

Автор: Abhyankar S.S.

Аннотация:

This book is a timely survey of much of the algebra developed during the last several centuries including its applications to algebraic geometry and its potential use in geometric modeling. The present volume makes an ideal textbook for an abstract algebra course, while the forthcoming sequel, "Lectures on Algebra II", will serve as a textbook for a linear algebra course. The author's fondness for algebraic geometry shows up in both volumes, and his recent preoccupation with the applications of group theory to the calculation of Galois groups is evident in the second volume which contains more local rings and more algebraic geometry. Both books are based on the author's lectures at Purdue University over the last few years.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2006

Количество страниц: 746

Добавлена в каталог: 28.03.2010

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Предметный указатель
Homogeneous localization (or algebraization of the geometry of projective varieties)      536—552 631
Homogeneous normalization      401—408 626
Homogeneous Normalization Theorem      402 626
Homogeneous polynomial      40
Homogeneous quotient field      536 631
Homogeneous radical ideal set      535
Homogeneous Resolution Lemma      436—441
Homogeneous resolutions      433—441 628
Homogeneous ring      213 612
Homogeneous Ring Normalization Theorem      408 627
Homogeneous submodules, characterizations and some properties (colons, radicals, primary decompositions, and associated primes)      394—395 399—400 624
Homogeneous syzygies      436
Homogeneously equivalent      436
Homogeneously finite free graded resolution      435
Homogeneously finite free module      434
Homogeneously finite free resolution      435
Homogeneously finite prefree graded resolution      435
Homogeneously finite prefree resolution      435
Homogeneously finite preprojective module      435
Homogeneously finite projective graded resolution      435
Homogeneously finite projective module      435
Homogeneously finite projective resolution      435
Homogeneously minimal free graded resolution      435
Homogeneously minimal free resolution      435
Homogeneously minimal prefree graded resolution      435
Homogeneously minimal prefree resolution      435
Homogenization      198
Homogenization (minimal)      536—540
Homogenize      67
Homomorphism      7—9 599—602
Homomorphism induced by      274 617
Homomorphism splits      445
Homothety      62
Homothety group      62
Hyperbola      30 63
Hyperboloid      63
Hyperplane      63
Hyperplane at infinity      67—68 550
Hyperquadric      63—70
Hypersurface      31 63—71 92—93 101—102 149
Hypersurface singularities      552
Hypersurface Singularity Theorem      552
Ideal      7 108 601
Ideal of an affine variety      148
Ideal set, nonunit ideal set, radical ideal set      115 148 373 609
Ideal-theoretic complete intersection (height equals generating number)      281 618
Ideal-theoretic direct sum      348
Ideally closed      415
Ideals And Homogeneous Ideals      594
Ideals And Homogeneous Ideals Relations Theorem      551 594
Ideals in a direct sum      341—343
Ideals in polynomial ring extensions      423
Ideals in polynomial rings      587
Ideals in rational function ring extensions      426
Ideals under field extensions      259
Ideals Under Transcendental Extension Theorem      417
Idempotent      341—343
Identify      37
identity map      3
Identity map of a set      445 584
Identity matrix      61
Identity permutation      24
Image      4 598
Immediate predecessor      375 623
Imperfect field      515 630
Impimitive (permutation) group      665
Implicit differentiation      64
Implicit function theorem      64
Impossible Real Ranks      686
Inclusion Relations Theorem      150 616
Inclusion Relations Theorem (Spectral and Projective versions)      262—264 544
Increase the roots      76
Independence Versus Regularity Theorem, and its Corollary      286—291
Independence, relative      236—241 614
Indeterminate      9—12 602
INDEX      5 600
Indexed family      108
Indexing set      46 108 599
Induced by (homomorphism)      273—274 617
Infimum      381
Infinite chain of prime ideals      250 615
Infinitely near points      161
Infinity (hyperplane or portion or points at infinity)      67—68 550
Inflexion      108
Initial form      92
Injection      3 598
Injective      3 584 598
Inseparability degree      642
Inseparability exponent      643
Inseparable element      515 630
Inseparable element (condition for)      646
Inseparable extension      515 630 635 649
Inseparable polynomials      515 630
Integral (extension of) over      78 161—163 241 606 614
Integral closure      78 161—163 606
Integral Closure Theorem (Finiteness version)      522
Integral dependence      78 161—163 606
Integral dependence (satisfaction)      171
Integral dependence in relation to irrelevant ideals      406 627
Integral element      78 161—163 606
Integral extensions and behavior of height and dimension (Dimension Corollary)      247 614
Integral extensions and preservation of fields and maximal ideals      243
Integral extensions and preservation under epimorphisms and localizations      242
Integral extensions and proper containment      245
Integral extensions and radical description      245
Integrally closed      78 161—163 606
Integrally graded ring      209
Internal direct sum of modules      202 611
Internal free additive abelian group      365
Intersection      4 598
Intersection multiplicity      65—66 108
Intersection of varieties      414—433 627
Intersection Theorem (Dimension of)      418 627
Intersections in vector spaces and projective spaces      433
Intersections of ideals or modules      108—109
Intuitive definition      149
Intval domain      80 81 188—192 606
Invariance lemma      447—452
Invariance-set      445
Invariant ideal      486
Invariant Ideal Lemma      486
Invariant partition      664
Inverse image      4 598
Inverse module      308 620
Inverse modules and fractional ideals      308
Invertible ideals and modules      355 621
Involution      305
Irreducibility      149
Irreducible affine variety set      147
Irreducible component      104 107 150 546
Irreducible element      13 602
Irreducible Ideal Characterization Lemma      304
Irreducible ideal decomposition      114
Irreducible ideal or submodule      109 608
Irreducible ideals and their theory      301
Irreducible projective variety set      542
Irreducible variety      104 147—148 535 542 548
Irredundancy      151
Irredundant premodel      156 633
Irredundant primary decomposition of ideals and submodules      114—117 609
Irrelevant Ideal Lemma      401 626
Irrelevant ideals      212—215 545—547
Irrelevant ideals in relation to integral dependence      406 626
Irrelevant submodule      400 626
Isobaric      101—102 180—181
Isolated component      118—119 170 609
Isolated positive upper segment      372
Isolated subgroup      56 372 623 681
Isolated subgroup (principal)      376
Isolated system of prime ideals of a submodule      400
Isolated system of prime ideals of an ideal      213 578
Isomorphism      7—9 152 599—602
Isomorphism Theorems and Groups      112
Isomorphism Theorems and Modules      111
Isomorphism Theorems First to Fifth      112
Iterated monoidal transform      570—571
Iwasawa's Simplicity Criterion      677
Jacobian      650
Jacobian Criterion of Separability      651
Jacobian matrix      650
Jacobson      98
Jacobson radical      219—220 261 613
Jacobson ring      261 616
Jacobson rings (with examples of rings which are not Jacobson)      583
Jacobson spectral variety      261
Jacobson spectrum      261 616
JORDAN      61
Jordan — Hoelder theorem      5 124—127
Kernel      7 599
Khahara      596
Klein      98
kronecker      32
Kronecker divisibility      581
Kronecker's theorem      83
Kroneckerian dimension      530
Krull      231 299 354 597 613
Krull intersection theorem      220—229 613
Krull Normality Lemma      355 622
Krull Normality Theorem      357 622
Krull's Struktursatz (Minimal Primes Theorem)      265 584 616
Kuttak Method of Ancient India as reported in Bhaskara's Beejganit      371
l'Hospital's rule      201
Lagrange      15
Lagrange Resolvent (Theorem)      15 636
Lagrange's theorem      651
Laplace development      90 172
Largest (element of a set of subsets)      301
Largest (or greatest or maximum) element in a poset      381
Largest proper subideal      302
Largest Subideal Characterization Lemma      302
Leading coefficient      393 606
Leading form      273—274 617
Leading ideal      273—274 617
Leading ideals and associated graded rings      585—587
Least (or smallest or minimum) element in a poset      381
Least common multiple      1
Least upper bound      381
Left regular representation      661
Leibnitz      59
Length of a chain      250 615
Length of a cycle      24—25
Length of a module      108 124—127 609
Length of a sequence or series      124—129
Length of an exact sequence      311 620
Lexicographic direct sum      377 683
Lexicographic disjoit sum      683
Lexicographic order      41 377 682
Lexicographic power      377 682
Lexicographic product      377 682
Lexicographic restricted power      683
Lie above      241—242 614
Lie below      241—242 614
Lies above      241—242 614
Lies below      241—242 614
Lifting homomorphism      442
Lifting property      442
LIMIT      53 86—89 608
Limit relative to      94—96 608
Limitations on Normalization Theorem      591
Limiting ordinal      34
Linear automorphism      92—94 98
Linear disjointness      414—433 627—628
Linear Disjointness (in various forms)      591
Linear Disjointness Lemma      417
Linear equations      162—165 174—177
Linear group      62—63 92 608
Linear independence      8—9
Linear map      9
Linear order      33
Linear transformation      62
Linearly disjoint      414—415 627—628
Linearly ordered set      33 603
lo      33
Local Analytic Geometry Book      364
Local complete intersection      301 619
Local ring      86 104 147—149 607
Local Ring Construction Theorem      146
Local rings (finite modules over)      393
localization      105 137—144 610
Localization at multiplicative set      137—144
Localization at prime ideal      144—146
Localization characterization      133 144
Localization ideal correspondence      139 145
Localization in the good case is a subring of the total quotient ring      138 144 242 610
Localization lemma      282
Localization of a module      330—339
Localization of Normality Lemma      242
Localization of regular sequences      291
Localization transitivity      139 144
Localizations (homogeneous and ordinary) to coincide      541
Localizations commute with epimorphisms, i.e., permutability of localizations and surjections      139 145
Localizations of direct sums      344—348
Locally free      442
Locally projective      442
Logical formality and narrative discourse      201
Long division      17
Loset      33 372—376 603
Loset of all nonempty segments      684
Lower segment (in a poset)      33 374
Lub      381
Lying Above Theorem      244 614
Lying Below Lemma      243
Macaulay      299
Map (it associates elements of a set called its domain to elements of a set called its range)      3—4 598
Mapping (it sends a specific element to its image)      482
Matrix      61
Matrix Lemma (First, Second, and Third)      513—514
Matrix of a homomorphism      446
Matrix ring      61
MAX      381
Maximal complementary prospectral variety      534
Maximal condition      11
Maximal element of a poset      33 381
Maximal ideal      7 17 46 601
Maximal ideals in polynomial rings      429
Maximal independent set      49 605
Maximal prospectral variety      534
Maximal prospectral variety set      534
Maximal prospectrum      534
Maximal regular sequence      280 618
Maximal spectral affine space      152
Maximal spectral projective space      548
Maximal spectral variety      115 147 609
Maximal spectrum      115 148 609
Maximal subgroup      665
Maximality ring      188—192
Maximality ring of a field      190—192
Maximum (or largest or greatest) element in a poset      381
Meet nicely or transversally      567
Mennike symbol      500 629
Mennike Symbol Lemma      502
Meromorphic series      34 604
Meromorphic series field      37 377 604
Meromorphic series ring      37 377 604
MIN      381
Minimal assassinator = colorful (like annihilator) long form of nass      116 216—217
Minimal associator      116 216—217
Minimal complementary prospectral variety      534
Minimal condition      129
Minimal element of a poset      381
1 2 3 4 5 6
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