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Jensen C.U., Lenzing H. — Model Theoretic Algebra with particular emphasis on Fields, Rings, Modules
Jensen C.U., Lenzing H. — Model Theoretic Algebra with particular emphasis on Fields, Rings, Modules

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Название: Model Theoretic Algebra with particular emphasis on Fields, Rings, Modules

Авторы: Jensen C.U., Lenzing H.

Аннотация:

It is the purpose of these notes to present some subjects from ring theory, field theory and module theory from a model theoretic point of view, basically, by making a semantic (first order) analysis of the corresponding algebraic concepts. Many non-trivial questions hereby arise, which may be of independent interest.


Язык: en

Рубрика: Математика/

Серия: Сделано в холле

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1989

Количество страниц: 447

Добавлена в каталог: 30.10.2011

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\lambda$-simple      199
$\Sigma$-algebraically compact indecompos-able module      245
$\Sigma$-algebraically compact module      160—161 163—164 169—171 193—195 216 245
$\Sigma$-algebraically compact ring      281
$\Sigma$-injective module      122 239 245
$\Sigma$-pure-injective module      161
${C}_{i}$-field      19 30—32
${\aleph}$-coherent ring      142 152 142
${\aleph}$-compact module      144 146 155 141
${\aleph}$-complete filter      140 152
${\aleph}$-generated module      136
${\aleph}$-injective functor      142—144 150—151
${\aleph}$-injective module      141
${\aleph}$-presented module      136
${\aleph}$-projective module      152
${\aleph}$-pure-exact sequence      139
${\aleph}$-pure-injective module      141
${\aleph}$-saturated      12
${\aleph}_{0}$-compact module      157
${\aleph}_{0}$-complete ultrafilter      156
${\aleph}_{0}$-incomplete ultrafilter      156
${\eta}_{1}$-ordered field      23 41 43
${\eta}_{1}$-set      23 173
${\mathcal F}$-product      140
(m,n)-presented module      228
Abelian category      381
Absolutely irreducible polynomial      6
Absolutely pure module      577
Additive category      379
Additive functor      319
Adic completion      118 190
Adic topology      118
Admissible field      77
Affine algebraic group      312
Affine group scheme      315
Affine k-algebra      310
Affine n-space      310
Affine scheme      313
Affine scheme, defined over the integers      316
Affine variety      310
Affine variety of modules      317
Algebraic compactness, of reduced power      154
Algebraic compactness, of ultrapower      287
Algebraic compactness, of ultraproduct      293
Algebraic number field      13 24 29
Algebraically closed field      5 13 16—17 19—21 23 30—31 40 43 49 58—59 306 309
Algebraically closed relatively      4
Algebraically compact module      125 128 130—131 144 154 157 165 171
Algebraically compact module, indecomposable      159 126 375
Algebraically compact module, over a Dedekind domain      205
Algebraically compact ring      281 285 287—288 323
Almost all      2
Almost-disjoint      195 175
Almost-split sequence      332
Amitsur's theorem      273
Amitsur-Levitzki's theorem      273
Anderson, F. W.      234 247
Annihilator of subfunctor      132
Archimedean field      41 48
Aronszajn, N.      207—208
Artin algebra      325 401
Artin-Schreier's theorem      21—23
Artinian module      401
Artinian ring      226 232 234—237 241 245 401
Atomic formula      307
Auslander's test      331
Auslander, M.      166 336 356
Auslander-Reiten quiver      336
Auslander-Reiten translate      333
Automorphism group of field      63
Ax and Kochen's theorem      31 69
Ax, J.      31 37 57
Ax-Kochen's theorem      31 60
Axiomatizability for Artinian rings      226 250
Axiomatizability for Ci-fields      32
Axiomatizability, for finite dimensional algebras      325
Axiomatizability, for finitistic projective dimension      263
Axiomatizability, for flat modules of rank n      244
Axiomatizability, for fp-injective modules      239
Axiomatizability, for fp-self-injective rings      278
Axiomatizability, for global dimension      262
Axiomatizability, for Goldie dimension      238
Axiomatizability, for infinite projective dimension      235
Axiomatizability, for infinite representation type      339
Axiomatizability, for n -generated flat modules      244
Axiomatizability, for self-injective dimension      278
Axiomatizability, for semifirs      256—257
Axiomatizability, for semihereditary rings      256—257
Axiomatizability, for semiperfect rings      252
Axiomatizability, for semiprimary rings      252
Axiomatizability, for uniformly coherent rings      254
Axiomatizability, for universally admissible fields      79
Axiomatizability, for weak global dimension      258—259
Axiomatizability, for weak global dimension one      256—257
Axiomatizable class      15—16 32
Axiomatizable class, of ${C}_{i}$-fields      19
Axiomatizable class, of algebraically closed fields      17
Axiomatizable class, of fields of characteristic zero      16
Axiomatizable class, of fields with trivial Brauer group      20
Axiomatizable class, of flat modules      122
Axiomatizable class, of formally real fields      21
Axiomatizable class, of Hilbertian fields      75
Axiomatizable class, of injective modules      122
Axiomatizable class, of Peano fields      61
Axiomatizable class, of Peano rings      61
Axiomatizable class, of projective modules      122
Axiomatizable class, of real closed fields      22
Axiomatizable class, of separably closed fields      17
Back-and-forth construction      39
Baer's test for injectivity      239 368
Baer, D.      126 163 209 245 293 297 299 356
Baer, R.      239 368
Barwise, J.      1 11 39 69 156—157 172 309
Base change      316
Bass's theorem      165—166 332
Bass, H.      122 165 232 234 251—252 263 271
Baur, W.      91—92 100 158 219
Bautista, R.      340—341
Bauval, A.      68 268
Bell, J. L.      1 5 12 16 20 61 159
Bergman, G.      268
Bernstein, I. N.      169 207
Bezout Domain      119 249 279 401
Bjork, J. -E.      251
Bongartz, K.      340 342
Boolean algebra free      261
Bound for coherence      229—230 232 254
Bounded representation type      165
Bourbaki, N.      52 114—115 229 231 257 305
Brauer group      20
Brauer group, trivial      20
Brauer-Thrall conjecture first      166
Brune, H.      291 297 299 356
Cancellation law, for function fields      30 35 57—58 70
Cancellation law, for polynomial rings      34 71 259 262
Cancellation law, for power series fields      30 56
Cancellation law, for power series rings      269—270 278
Cartan, H.      123 231 234 256
Cassels, J. W. S.      25—26 28
Category, abelian      381
Category, additive      379
Category, of additive functors      384
Category, path      405
Chang, C. C.      1 11—12 15 24 31 46 57 69
Characteristic transfer principle      6
Chase, S. U.      122 163 229 260
Cherlin, G. L.      23
Chevalley's Theorem      311 317
Chevalley, C.      313
Class, axiomatizable      15—16
Class, elementarily closed      13
Class, finitely axiomatizable      15
Closed subscheme      314 345
Cogenerator      401
Cohen, I. S.      232
Coherent module      237
Coherent ring      97 99 106 108 122 124 229 231—232 236—237 253 255 258—259 264—265 278—279 377 401
Cohn, P. M.      184 257
Colliot-Thelene, J. L.      28
Commutative Noetherian ring      113 115—122
Compact module      158
Compactness Theorem      3 16
Complete local ring      56
Completion-adic      29 59
Complex number field      23
Constructible set      311
Constructible subscheme      315 342
Continuum Hypothesis      24 172 293—294
Continuum hypothesis, generalized      12
Coordinate algebra      310
Coordinate ring      313
Countable ring      157 235—236 239
Crawley-Johnsson-Warfield theorem      234
Cubically closed field      55
Curtis, C. W.      324
Cyclically presented module      97
Dedekind domain      98 109 115 168 170 182 190 201 216 218 232 250—251 265 291 402
Dedekind, R.      249
Definable subfunctor      135
Definable subgroup      125 161 171
Delon, F.      57 60 74
Demazure, M.      313
Dense functor      380
Descending chain condition, for closed subsets      311
Descending chain condition, for submodules      401
Descending chain condition, on finitely generated ideals      251
Descending chain condition, on principal ideals      235 251
Descent infinite      35 51
Diagonal embedding      2 4
Diagonal mapping      2 125
Diagram, expansion      308
Diagram, language      308
Dieudonne, J.      207
Dimension, embedding      265
Dimension, finitistic global      313
Dimension, flat      369
Dimension, fp-injective      264 377
Dimension, Gelfand-Kirillov      275 277—278
Dimension, global      278 369
Dimension, Goldie      238
Dimension, injective      369
Dimension, Krull      117—118 121 123 254 271 276—278 313 378
Dimension, local      313
Dimension, of topological'space      272
Dimension, projective      234 369
Dimension, pure-global      155 291—292 375
Dimension, pure-injective      155 375
Dimension, pure-projective      291—292
Dimension, weak global      370
Dimension, weak homological      231 369
Discrete valuation ring      56
Dlab, V.      209
Domain, factorial      50 378
Domain, principal ideal      406
Domanov, O. I.      268
Douady, A.      77
Dries, L. van den      29 77 307
Dugas, M.      63
Dynkin diagram      169
Eakin, P.      270
Eilenberg, S.      123 231 234 256
Eklof, P. C.      37 122
Elementarily closed class      13
Elementarily closed class, of algebraically closed fields      13
Elementarily closed class, of flat modules      122
Elementarily closed class, of projective modules      122
Elementary class of representation-finite rings      301
Elementary definability      33
Elementary definability, in field of quotients      61—62
Elementary definability, in number field      66
Elementary definability, of $K[[{X}_{1},...,{X}_{n}]]\;in\;K\;(({X}_{1},...,{X}_{n}))$      50
Elementary definability, of field in function field      34
Elementary definability, of Henselian ring in quotient field      50 56
Elementary definability, of integers in polynomial ring      36
Elementary definability, of Q in polynomial ring      36
Elementary definability, of ring in field of quotients      56
Elementary descent      229—230 235 237 240 244
Elementary descent, for $\Sigma$-algebraically compact modules      239
Elementary descent, for $\Sigma$-algebraically compact rings      287
Elementary descent, for $\Sigma$-injective modules      239
Elementary descent, for algebraically compact modules      239
Elementary descent, for algebraically compact rings      285
Elementary descent, for Artinian modules      237
Elementary descent, for Artinian rings      237
Elementary descent, for coherent rings      255
Elementary descent, for Dedekind domains      251
Elementary descent, for flat modules      229
Elementary descent, for free modules      235
Elementary descent, for GK-dimension      276—277
Elementary descent, for global dimension      260 262
Elementary descent, for injective modules      239—240
Elementary descent, for Krull dimension      276
Elementary descent, for locally coherent modules      237
Elementary descent, for locally coherent rings      237
Elementary descent, for maximum condition      244
Elementary descent, for minimum condition      244
Elementary descent, for Noetherian modules      237
Elementary descent, for Noetherian rings      237 251
Elementary descent, for perfect rings      253
Elementary descent, for primitive rings      268
Elementary descent, for principal ideal domains      251
Elementary descent, for projective modules      235
Elementary descent, for pure-projective modules      235
Elementary descent, for regular local rings      262
Elementary descent, for transcendency degree      277
Elementary descent, for weak global dimension      259
Elementary embedding      4
Elementary equivalence      11 15
Elementary equivalence of fields      32
Elementary equivalence of Henselian fields      31
Elementary equivalence of power series fields      31
Elementary equivalence, for algebraically closed fields      5
Elementary equivalence, for embedding dimension      266
Elementary equivalence, for fields      13—14
Elementary equivalence, for flat modules      106 115—119 229
Elementary equivalence, for fp-injective dimension      240
Elementary equivalence, for fp-injective modules      108
Elementary equivalence, for fp-self-injective rings      265
Elementary equivalence, for free algebras      266
Elementary equivalence, for function fields      14 28 30 43 47—49 57—58 70
Elementary equivalence, for global dimension      261
Elementary equivalence, for injective dimension      240
Elementary equivalence, for injective modules      120
Elementary equivalence, for injective modules over commutative Noetherian rings      121
Elementary equivalence, for Krull dimension      276
Elementary equivalence, for localizations      112—113
Elementary equivalence, for matrix rings      238 274
Elementary equivalence, for modules over Z[X]      115
Elementary equivalence, for Noetherian rings      251
Elementary equivalence, for number fields      13
Elementary equivalence, for polynomial rings      36—37 39 43 71 259 262
Elementary equivalence, for power series fields      31 56—57
Elementary equivalence, for power series fields $K(({X}_{1}))...(({X}_{n}))$      30
Elementary equivalence, for power series fields $K(({X}_{1}...{X}_{n}))$      31
Elementary equivalence, for power series rings      43 269—270 278
Elementary equivalence, for R-modules      100 107
Elementary equivalence, for rational function fields      28
Elementary equivalence, for real closed fields      9 23 41—42
Elementary equivalence, for rings of continuous functions      272
Elementary equivalence, for self-injective rings      265
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