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Cohn P.M. — Free Rings and Their Relations (London Mathematical Society Monographs)
Cohn P.M. — Free Rings and Their Relations (London Mathematical Society Monographs)

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Название: Free Rings and Their Relations (London Mathematical Society Monographs)

Автор: Cohn P.M.

Аннотация:

In the development of ring theory there are two strands, each with its own problems and methods, and although with many points of contact, they have never merged completely. One of them is the theory of algebras; this was a non-commutative (indeed sometimes even non-associative) theory from the beginning but there were always heavy finiteness restrictions, which are only gradually being relaxed. Thus while there is a fairly substantial theory of Artinian rings, the theory of Noetherian rings is still in its early stages, and although it is being developed vigorously, it is clear that some type of maximum condition is essential for its development.
A second and quite distinct line originated with the study of arithmetic in algebraic number fields. In the hands of (Cummer, Dedekind and E. Noether this led to the abstract notion of a Dedekind ring. Meanwhile, algebraic geometers found the need for affine rings hi the study of algebraic varieties; here Dedekind rings appear again, but as a rather special case (essentially the l-dimensional case). Now in the last few years our way of describing the geometrical notions has changed quite radically, with the result that the correspondence (rings)—>(varieties) has been extended and made precise: There is a contravariant functor from the category of commutative rings to the category of affine schemes, which is an equivalence. The effect of this connexion has been profound in both directions; we are concerned here particularly with the help afforded by the geometrical notions in studying rings...


Язык: en

Рубрика: Математика/

Серия: Сделано в холле

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2nd edition

Год издания: 1985

Количество страниц: 608

Добавлена в каталог: 15.05.2011

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$ACC_{ds}$ = ACC on finitely generated submodules with dense inclusions      289
$ACC_{n}$ = ACC on n-generator sub-modules      5 71f. 287
$GE_{2}$, standard form      115f.
$GL_{n}$ = general linear group      xviii
$K_{1}$      449
$Lat_{R}(M)$ = lattice of submodules of M      3
$Tr_{n}$ = group of triangular matrices      74
$ГЉ_{0}$      19 63
$\alpha$-adic filtration      125
$\Sigma$-inverting homomorphism      382
$\Sigma$-inverting ring (universal)      390
$\Sigma$-rational closure      382
Abelian category      546
Abelianizing a ring (or group)      341 448f.
Absolute property      205
Abstract atomic factor      215
ACC = ascending chain condition      5 154
Additive category      54
Additive functor      549
Adjoint Associativity      552
Adjoint functor, pair      548
Admissible system, matrix      384 421
Affine automorphism      342
Affine scheme      487
Algebraic algebra      220
Algebraic element (left, right)      515
Algebraic matrix      506
Algebraic power series      134 291
Amitsur's theorem (on generalized polynomial identities)      281ff.
Anti-ideal      322 334 376 379
Artin's problem      521
Artinian ring = ring with (left, right) DCC      22
ASSOCIATED      25 153
Atom, (n-) atomic      73 154 168
Augmentation ideal      61
Augmentation preserving automorphism      61 342
Baer's criterion      80 549
Basis theorem for abelian groups      494
Bergman — Dicks localization theorem      459
Bergman's centralizer theorem      340 378
Bergman's conjugacy theorem      529
Bezout domain (left, right)      12 69
Bezout identity      86
Bezout ring (weak)      86
Biased      357
Biassociated      422
Bicentralizer      282
Bidegree      344
Binomial extension      518
BirkhofTs representation theorem      180 544
Bordered matrix      405 421
Bound component, module      228f.
Bound, strongly      234
Bounded element, module      312 379
Bounded linear transformation, matrix      505
Cancellation monoid      35
Capacity      22
Card(I) = cardinal of I, also |I|      xx
Category      544
Cayley — Hamilton theorem      505 539
Central extension      518
Chain ring      170
Characteristic of a module      24
Chase's lemma      175
Cleavage, cleft      189
Closed submodule, closure      245 251
Closed subring      102
Code      324 379
Cofinal      73 289
Cohen's theorem      310
Coherent family of matrix groups      74
Coherent ring      258 554
Coinduced extension      552
Cokernel      545
Column rank      247
Comaximal relation      28 161
Comaximal transposition      178 270
Comaximally transposable      171
Comma category      548
Comma-free code      379
Commensurable      xix
Companion matrix      505
Comparison theorem for numerators and denominators      429
Complete direct decomposition      186
Complete inversely filtered ring      126
Complete lattice      540
Completely primary ring      187
Completely reducible module      189
Completion (of a filtered ring)      126
Complex-skew polynomial ring      54 164 179f. 191 210 225
Computable matrix      15
Conductor      299 304
Conical monoid      19 153 321
conjugate      504
Connected (inversely filtered ring)      129
Connecting homomorphism      550
Constants      40
Content      107
Continuant polynomial      117
Convex      219 526
Coprime relation      161
Core      384
Cover (in a lattice)      540
Cramer's rule      384
Cyclic matrix      504 539
DCC = descending chain condition      10
de Jonquieres automorphism      342 379
Decomposable (left, right)      184 316
Dedekind lemma      372f.
Defect theorem      329
Degenerate matrix      400
Degree      60 100 111 344 349
Degree of filtration      95
Degree, formal      104
Degree-function      51 139
Denominator      36 384
Dense subcategory      314
Dense submodule      245
Dependence number, $\lambda_{\nu}(-)$      96 151
Dependence number, inverse, $\mu_{\nu}(-)$      125
Dependence relation      64 100 475
Dependence relative to a filtration      95
Dependence relative to a transfinite function      139
Depth      386 432ff.
Derivation      39
Derivation, higher      137 358 523
Derived set      88 151
Determinantal sum      396
DFL = distributive factor lattice      199 207
Diagonal reduction      489 538
Diagonal sum      xix 395
Diamond lemma      148
Dicks commutator test      355f.
Dieudonne determinant      382 448
Differential equations      63
Differential operators      54 58
Differential transformations      503
Dimension, injective, projective      258 551
Direct product      544
Direct sum      545
Directed set      292 398
Distributive factor lattice, DFL      199
Distributive module      193
Distributiye lattice      542
Division algorithm      87ff.
Division closure      335 387
Divisor group, D(-)      451
DL = category of finite distributive lattices      211
Duality = category anti-equivalence      548
Duality factorial      166
Duality for modules      xviii 165 232ff.
Duality in lattices      540
E-related      119
Eigenring, E(-)      30 63 192
Eigenring, scalar      220
Eigenvalue      504 510
Elementary divisor      495
Elementary divisor ring      492 539
Elementary embedding      287
Elementary matrix, operation      490
End of a graph      352
Epic R-field      388
Epic, epimorphism      545
Equivalence of categories      548
Equivalence of factors      215
Essential extension      235
Essential left factor      243
Euclidean algorithm      89 121f. 151
Euclidean ring      88
Euler's theorem (homogeneous functions)      62
Exact functor (left, right)      549
Exact sequence      546
Exchange Principle      126
Ext, extension of modules      551
Extremity of a graph      351
Factor closed      415 454
Factor complete      454
Factorial duality      166
Fibonacci numbers      117
Field of fractions      38 251 388
Field spectrum, X(-)      410 487
Filtered ring      95
Filtered ring, truncated      111
Filtration      94
Filtration, $\alpha$-adic      125
Final object      545
Finitely generated (presented, related)      24
Fir (left, right), n-fir      67 71 85
Fir, one-sided      142 176 536
Firoid      86
Fitting's lemma      187 197 240
Five-lemma      547
Fixed ring      362
Flat module      49 552
Formal degree      104
Formal power series      55
Fox derivative      55
Fractions (monoid, group, ring, field)      34ff.
Free algebra      59 106
Free ideal ring ( = fir)      71
Free monoid      59
Free product of groups      349
Frobenius inequality      261
Full matrix      159 192
Full operation      422
Full subcategory      545
Fully atomic      168
Fully invariant      195
Fully inverting homomorphism      415
Fully reducible      189 314
functor      547
Galois correspondence for free algebras      376ff. 380
Gauss's lemma      85f. 107 124
GE-related      119
GE-ring      76 86
Generalized polynomial identity, GPI      282
Gerasimov — Malcolmson localization theorem      484
GL-related      119 162
Global dimension      551
Global section (rational, integral)      487
Graded ring      100
Grothendieck category      239
Group of fractions      34
Hasse's criterion      94
HCF, HCLF, HCRF = highest common (left, right) factor (=greatest common divisor)      122 150 154
Height (of lattice element)      542
Hereditary ring      11
Hermitering      15 63
Higher derivation      137 358 523
Higman's trick      272
Hilbert basis theorem      53 63
Hilbert series      107 364ff.
Hilbert theorem      90 521
HNN-construction      114 151f. 421
Hollow matrix      160 397
Homogeneous      345
Honest homomorphism      250 415
I-atom, I-prime      156 301
I-decomposable      316
IBN = invariant basis number      6 63
Idealizer, I(-)      30 63
Indecomposable      186 316
Index of a matrix, i(-)      28
Inert (totally, n-)      83 86
Inertia theorem      133 152
Initial object      545
Injective dimension      551
Injective hull      549
Inner derivation      40
Inner rank      248 487
Integral element, closure      298f. 303
Intersection theorem for firs      295 297
interval      540
Invariant      362 379
Invariant basis number (IBN)      6 63
Invariant elements, I(-)      154 156 300 379
Invariant factors      495 504
Invariant ring (monoid)      154 156
Inverse dependence number, $\mu_{\nu}(-)$      125
Inverse filtration      125
Inverse weak algorithm (n-term)      125
Inverting      34 382
Involution      294
Irreducible topological space      411
Irredundant decomposition      411 543
Isomorphism in a category      545
Isomorphism of factorizations      164
Isomorphism of idempotent matrices      20
Isotone (=order-preserving) mapping      211
Iterated skew polynomial ring      532
J-ring      532
Jacobian matrix, problem      355 380
Jacobson radical, J(-)      20
Join      540 543
Join-irreducible      212 543
Jordan — Hoelder theorem      168 198 214 542
Jung-van der Kulk theorem      348 379
Kaplansky's theorem (projective modules)      12 63
Kernel      45 545
Kharchenko — Galois correspondence      376
Kharchenko — Lane theorem      363f. 380
Klein's theorem      82 86
Kraft — McMillan inequality      325 379
Kronecker functional ring      308
Krull domain      302
Krull — Schmidt theorem      186 241 297 494 543
Kurosh — Ore theorem      185 543
L(cR, R) = lattice of principal right ideals between cR and R      80 179
Lane's lemma      357 380
Large (left, right)      36
Laurent series      56 521ff.
LCM, LCRM, LCLM = least common (right, left) multiple      81 154
Leading form      344
Leading term      101 140 527
Leapfrog construction      117
Leibniz's formula      46
Length of a lattice      542
Length of an element, monomial      60 322 494
Level      174
Lifting property      77
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