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Cohn P.M. — Skew Fields : Theory of General Division Rings (Encyclopedia of Mathematics and its Applications)
Cohn P.M. — Skew Fields : Theory of General Division Rings (Encyclopedia of Mathematics and its Applications)



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Название: Skew Fields : Theory of General Division Rings (Encyclopedia of Mathematics and its Applications)

Автор: Cohn P.M.

Аннотация:

Algebraists have studied noncommutative fields (also called skew fields or division rings) less thoroughly than their commutative counterparts. Most existing accounts have been confined to division algebras, i.e. skew fields that are finite dimensional over their center. This work offers the first comprehensive account of skew fields. It is based on the author's LMS Lecture Note Volume "Skew Field Constructions". The axiomatic foundation and a precise description of the embedding problem precedes an account of algebraic and topological construction methods. The author presents his general embedding theory with full proofs, leading to the construction of skew fields. The author has simplified his treatment of equations over skew fields and has extended it by the use of matrix methods. A separate chapter describes valuations and orderings on skew fields, with a construction applicable to free fields. Numerous exercises test the reader's understanding, presenting further aspects and open problems in concise form. Notes and comments at the end of chapters provide historical background. The book will appeal to researchers in algebra, logic, and algebraic geometry, as well as graduate students in these fields.


Язык: en

Рубрика: Математика/Теория чисел/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1995

Количество страниц: 258

Добавлена в каталог: 28.05.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
-adic topology      66
-adic valuation      85 425
-algebra      41
-ring      41 335
Abelian valuation      427
Abelianization      437
Absolute GRI      340
Absolute presentation      317
Absolute value      422
Abstract support system      362
AC = algebraically closed      308
Addition      3
Additive group of a field      4
Admissible matrix block, system      159
Algebraic closure      308 329
Algebraic dependence      406
Algebraic element      111
Algebraic extension      140
Algebraic matrix      379
Algebraically closed      308 367
Amalgamation property      310
Amitsur — Levitzki theorem      334 343
Amitsur’s GPI-theorem      332 365
Amitsur’s theorem on rational identities      339 344
Annihilator ring      393
Antichain      72
Artin — Schreier condition      461
Artin’s theorem      101 103f
Associated matrices      24
atom      27
Augmentation ideal      44
Augmentation mapping      55
Augmented ring      44
Base ring in a coproduct      211
Basic formula      309
Basic module      211
Bezout Domain      13
Bicentralizer      110
Binomial field extension      121
Birkhoff — Witt Theorem      89
Block      162
Bokut’ classification      330
Bound module      34
Bound of an element      28
Bounded element      28
BW-algebra      90
CAC = characteristically algebraically closed      370 418
Cartan — Brauer — Hua Theorem      144
Cayley — Hamilton theorem      385
Central      62
Central field extension      121
Central localization      387
centralizer      4
Centralizing extension      339
Centre      4
Characteristic of a field      4
Che valley’s extension lemma      426 470
Cofinite subset      11
Cogenerator      389
Comaximal      25
Comma category      203
Commutator      143 238 259 441
Compactness theorem of logic      327 329
Companion matrix      370
Complete variety      416
Concatenation of valuations      433
Cone      458
Conical monoid      73 249
conjugate      112 233
Consistent system of equations      309
Convex subgroup      432
Coprime      26 29
Coproduct in a category      203f
Coproduct of fields      223 268 449
Coproduct of rings      42 222 276
Core      159
Cramer’s Rule      160
Critically skew field      149
Crossed product      123ff
Cyclic extension      133
Cyclic matrix      381
Dedekind’s lemma      101f
Defect of a matrix over $D_k\langle{x}\rangle$      409
Degenerate      336 344
Degree of a field extension      94
Degree of a monomial      215
Degree of a polynomial      48 84
Denominator of an admissible matrix block      162
Denominator of an admissible system      159 201
Dependable field extension      318
Dependence relation      406f
Depth      196
Derivation      49
Derived group      144 427
Desarguesian plane      1 365
Determinantal sum of matrices      163
Diagonally closed matrix set      445
Dieudonng determinant      437 466
Differential equation      133
Differential polynomial      46 91
Distributive laws      4
Divergence of a skew polynomial      59
Divisibility ordering      73
Division algebra      5 45 150
Division ring      4
Domain of a function      336 344
Domain of a subvaluation      444
Dominate      428
E-associated      436
E-matrix      437
EC-field      309ff
Effective construction      318
Eigenring      28
Eigenspace      378
Eigenvalue      370 375 378 403
Eigenvector      375
Elementary divisor      382
Elementary mapping      311
Elementary sentence      308
Elimination theory      416f
Embedding condition      326
Epi-final, -initial      177
Epic-field      154 336
Epimorphism      153
Equivalence of homomorphisms      101
Equivalence of local homomorphisms      154
Equivalence of matrix blocks      163
Equivalence of places, valuations      424
Essential extension      387
Essential index (term)      349
Essential set      354
Euclidean algorithm      79
Evaluation map      333
Existential sentence      308
Existentially closed      309ff
FAC = fully algebraically closed      371
Factor ring in a coproduct      211
Factorial duality      27
Faithful A-ring      41 205
Faithful coproduct      42 204
Field      3 154 336
Field coproduct      223
Field of fractions      8
Field, spectrum      173 200
Filter      11
Filtered ring      83 228
Filtration      83 206f
Finite extension      94
Finite field element      404
Finite topology      98
Finitely generated      280
Finitely inert      284
Finitely presented      280
Finitely presented, point      417
Finitely related      280
FIR      36f 40 46 222 248
Flat module      228
Flat subset      337
Forcing companion      311
Formally real field, ring      459
FRACTION      15
Free fc-algebra      224
Free field      224 235 301 344
Free ideal ring      36f see
Free module type      21
Free point      402
Free product of groups, rings      42 204f
Free product with amalgamation      275
Free set in a field      235 279f
Free transfer isomorphism      220
Full map      193
Full matrix      22 168 179
Fully algebraically closed      200 371
Fully inverting map      177
Function field      338
Fundamental theorem of Galois theory for skew fields      109
Galois connexion      99 109 142
Galois extension, group      100 129ff
Generic division algebra      344
Generic field      329
Generic matrix ring      46 343
Generic point      414
Gerasimov — Malcolmson localization theorem      40
Global dimension      39 218 244 247
GPI = generalized polynomial identity      283 332 339
Graded ring      83
GRI = generalized rational identity      337 340f
Grothendieck group of projectives      186
Hahn — Banach theorem      18 420 442
HCLF = highest common left factor      32
Height      207
Height, theorem for integral domains      208
Hereditary ring      39 246
Hermite ring      457
Higman’s theorem      316 325
Higman’s trick      284
Hilbert basis theorem      90
Hilbert field      281 339 469
Hilbert nullstellensatz      411ff 415 419
Hilbert “Theorem 90”      135f
HNN-extension of a field      231f 276
HNN-extension of a ring      239ff
Hochster’s axioms      200
Hollow matrix      179 299
Homogeneous element      84
Homogeneous field      233
Homological dimension      39 214
Honest homomorphism      177
Horn sentence      9
Hua’s identity      335
Hua’s theorem      144 150
I-atom      29
IBN = invariant basis number      19 46 249 328
Idealizer      28
Idempotent      184f
Identity      9 331f 335
Indecomposable      31
Index of a matrix      24 296
Induced homomorphism      219
Induced module      186 211
Inductive class      311
Inertia lemma      284
Inner derivation      50
Inner eigenvalue      378
Inner Galois group      109
Inner order of an automorphism      61
Inner rank      179 192
Integral domain      8
Internal modification      35
Invariant basis number      19
Invariant element      28 57f
Invariant factor      380 384
Invariant subring      423
Inverse eigenvalue      403
Inversive ring      33
Inverting homomorphism      14 156
Involution      56
Irreducible algebraic set      337
Irreducible element      27
Isomorphic idempotents      185
Iterated skew polynomial ring      78ff
J-ring, J-skew polynomial ring      79
Jacobson radical      10 82 92
Jacobson — Bourbaki correspondence      99 150
Jacobson — Zassenhaus formula      137
Jategaonkar’s condition      79
Jordan matrix, normal form      384
Kaplansky’s Pi-theorem      332 365
Kaplansky’s theorem on projective modules      191
Key term      216
Klein’s nilpotence condition      23 40 250
Klein’s theorem      23 40
Large submodule      387
Laurent polynomial      44 55
Laurent series      45 66
LCLM, LCRM, least common left, right, multiple      33
Leading term      84 215
Length of an element in a UFD      28
Level of a field      470
Level of an affine space      403
Lie algebra      88
Linear companion      294 370
Linear matrix      293
Linearization by enlargement      284
Linearly disjoint      96 302
Local homomorphism      154
Local ring      154 169 344
localization      15 156
Locally cyclic group      146
Locally finite algebra      387
Locus of a point      403
Lower central series      76
Magic lemma      438
Malcev conditions      9 23
Malcev — Neumann construction      76 79 91
Malcolmson’s criterion      167 183
Matrix block      159
Matrix cone      463
Matrix functor      42
Matrix ideal      172
Matrix local ring      344
Matrix preideal      173
Matrix reduction functor      43 46 247
Matrix subvaluation      443
Matrix units      42
Matrix valuation      435
Matrix-algebraic algebra      387
Matrix-homogeneous      234 391
Metacyclic group      148
Metro-equation      369 418
Minimal invariant element      59
Minimal polynomial      113
Monic matrix      293
Monic polynomial      48
Monoid of projectives      186 221 244 249
Monoid ring      55
Monomial matrix      291
Monomial unit      208
Multiplication      3
Multiplication algebra      258
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