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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
Linear automorphism      342
Linear differential operator      191 506
Linearization by enlargement      272
Link in a lattice      180
Local homomorphism      388
Local rank      255
Local ring (scalar, matrix)      22
localization      37 81 390
Localization theorems      457ff. 472 484
Locally free module      48
Lower multiplicative set      382
Lower segment      211 333
Magnus'theorem      296
Makar — Limanov — Czerniakiewicz theorem      355 379
Malcev conditions      35 82
Malcev — Neumann construction      528
Malcolmson's criterion      485
Matrix (pre-) ideal      396ff.
Matrix algebraic      220
Matrix basis      1
Matrix local ring      22
Matrix object      463
Matrix reduction functor, $\mathfrak{W}_{n}$      147f. 152
Maximal code      325
Meet      540
Meta (semi-) fir      70 73
Metro-equation      515
Minimal admissible matrix      428
Minimal factorization      248
Mixed free algebra      62
Modular lattice, law      540
Monic matrix over a free algebra      272
Monomial      60
Monomial term      105
Monomorphism      545
Morita equivalence      4 62 78
Multiplicative set      382
Nagata's formula      364
Nagata's theorem      157
Nakayama's lemma      21
Natural isomorphism, transformation      548
Negative module, Neg      236 297
Noetherian ring = ring with (left, right) ACC      15
Non-singular matrix      xix
Normal closure      443
Normal form in a free algebra      59
Normalizes      443
Null matrix      3
Nullity      161
Numerator      384
One-sided fir      142 176 536
Opposite category      544
Order      55 60 421
Ordered series ring      527
Ore condition, domain      38
Ore set      35
Outer derivation      40
Outer rank      252
Parallelogram law      541
Partition lemma      67
Persistent property      205 442
Perspective      541
Place permutation      364
Polycyclic module      244
Pontryagin's theorem      297
Pos = category of finite partially ordered sets      211
Positive module, Pos      236 297
Power series D-ring      61
Preadditive category      545
Prefix set, code      324
Presentation      552
Primal ring      252
Primary decomposition      184
Primary element      191
Prime      449
Prime element      154 156
Prime matrix      160 425
Prime matrix ideal      401
Prime module      236
Principal ideal domain, ring, PID      71
Principal valuation ring, PVR      130 173
Product of matrix ideals      400
Projective dimension      551
Projective free ring      17 63
Projective module      10 549
Projective trivial ring      18
Projective, in a lattice      541
Proper factorization      268
Proper matrix relation      161 266
Proper specialization      393
Protorsion module      239
Pseudo-Sylvester domain      263 297 418
Pseudo-valuation      95
Pseudolinear extension      517
Pseudolinear mapping      503
Pullback      545
Pure extension      518
Pure subalgebra      341
Pushout      545
PVR = principal valuation ring      130 173
Quasi-Frobenius ring      304
Quasi-identities      414 419 486
Quasi-variety      419 486
Quaternions      23 93 158 225f. 514
Quotient monoid      154
R-field (universal)      388f.
Radical of a matrix ideal      402
Radical of a matrix ideal, Jacobson      20 171
Radical of a matrix ideal, tertiary      321
Rank formula (modules over semifirs)      68
Rank of a free algebra      60
Rank of a free module      6
Rank of a matrix      245ff. 424 431
Rank of a module      48 251
Rational closure      382
Rational power series      134
Rational section      487
Rationality criterion      526
Reduced admissible matrix      429
Refinement of a factorization      178
Regular      xix
Regularly embedded      330
Relation (n-term, trivial)      64 100
Relevant N-value      460
Resolution (injective, projective)      258 551
Retract of a ring      70 221
Right invariant      203 301
Rigid domain, UFD      170 192
Rigid element      41 170
Rigid factorization      190
Rigid monoid      41
Ring of fractions      37
Ringlike matrix object      463
Ringoid      86 545
Root      323 339
Row rank      247
S-ring      263 418
Saturated set, saturation      394 467
SAut, Special automorphism group      357
Schanuel's lemma      24f. 192 552
Schreier refinement theorem      178 542
Schreier — Lewin formula      109 334
Schreierset      333
Schur's lemma      167 222
Section, global (integral, rational)      487
Segment (lower, upper)      211
Semi-Artinian, -Noetherian      195
Semifir      67 85
Semifree module      48
Semihereditary (left, right, weakly)      11 13 63
Semilinear mapping      503
Semiprime matrix ideal      402
Semisimple Artinian ring      193 550
Shear      342
Similar elements, matrices, $\sim$      25 162 192
Similar right ideals      158
Simple N-value      450
Simple, $\mathcal{F}$-simple      167
Singular kernel      391
Singular matrix      xix
Singularity support      286 410
Skew formal Laurent series      522f.
Skew Laurent polynomial      54
Skew polynomial ring      52 497ff.
Skew polynomial ring, iterated      532
Skew rational function field      53 521
Source      544
Spacial module      254
Special automorphism      357
Specialization      388
Specialization lemma      285 297
Spectral space      412
Spectrum of a matrix      510
Split exact      549
Squarefree module      196
Stabilize, stab      342
Stable atom      17
Stable GL      449
Stable rank      261
Stably associated      26 63 162
Stably free      15
Stably full      262
Stably honest      418
Standard basis in $R^{n}$, $ ^{n}R$      3
Standard form in $GE_{2}$      115f.
Strictly bordered      426
Strictly cyclic module      192
Strong DFL-property      201 299f.
Strong G-ring, $G_{n}$-ring      76
Strong prime ideal      177
Strongly regular      419
Subdirect product      544
Subdirectly irreducible      544
Suffix set      324
Support      59 354 526
Sylvester domain      253 297 417
Sylvester's law of nullity      253 297
Tame automorphism      343
TARGET      544
Tensor D-ring (on a D-bimodule)      61
Tensor product of modules      552
Tertiary radical      321
Test module      192
Three-by-three lemma      547
Top component      344
Topological fir      128
Torsion class, torsion free class      228 296
Torsion element, submodule      46
Torsion free module      46
Torsion module (over semifir)      164 239 297
Total divisor      489
Totally coprime      315 435
Totally inert      84
Totally unbounded      316
Trace form      372
Transcendental matrix      506
Transduction      105
Transfinite degree function      139
Transfinite division algorithm      90
Transfinite weak algorithm      139 152
Translation      342
Transpose of a module, Tr(-)      233 297
Transvection      74
TREE      349
Trivial filtration      95
Trivial operation      405 422 469
Trivial relation      64 475
Trivializable      65 101 475
Truncated filtered ring      111
U(R) = group of units of R      xviii
UF-monoid      153f.
UFD = unique factorization domain      153 164 192
UGN (= unbounded generating number)      6 63
Ultraproduct theorem      286
Unbound      228
Unbounded generating number (UGN)      6 63
Unbounded, totally      316
Uncleft, totally      189
Unfactorable matrix      269
Unique factorization domain (UFD)      153 164 192
Unique remainder algorithm      90 532 539
Unitriangular matrix      74
Universal denominators, theorem on      434
Universal field of fractions, R-field      281 389
Universal group of a monoid      35
Universal localization      390
Universal mapping property of free algebras      59
Universal object      548
Universal T-inverting monoid, ring      34 36
Upper multiplicative set      382
Valuation (ring)      302 452 531
Value      445
Vandermonde's identity      360
Weak $\nu$-basis of a right ideal      98 128
Weak algebra basis      104 129
Weak algorithm (n-term)      96 273
Weak algorithm (n-term) in a graded ring      101f.
Weak algorithm (n-term), inverse      125
Weak algorithm, transfinite      139
Weak global dimension      256ff. 553
Weakly finite ring      6 63
Weakly semihereditary      13
Weight      342f.
Weyl algebra, $A_{1}$      55 265
Whitehead group      449
Whitehead lemma      451 453
Width of a graph      351
Wild automorphism      342
X-inner, X-outer automorphism      372
Zero (left, right) of a polynomial      514
Zero delay code      324
Zero object      545
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