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Anderson F.W., Fuller K.R. — Rings and Categories of Modules
Anderson F.W., Fuller K.R. — Rings and Categories of Modules



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Название: Rings and Categories of Modules

Авторы: Anderson F.W., Fuller K.R.

Аннотация:

This book is intended to provide a self-contained account of much of the theory of rings and modules. The theme of the text throughout is the relationship between the one-sided ideal structure a ring may possess and the behavior of its categories of modules. Following a brief outline of the foundations, the book begins with the basic definitions and properties of rings, modules and homomorphisms. The remainder of the text gives comprehensive treatments of direct sums, finiteness conditions, the Wedderburn-Artin Theorem, the Jacobson radical, the hom and tensor functions, Morita equivalence and duality, decomposition theory, and semiperfect and perfect rings. This second edition includes a chapter containing many of the classical results on Artinian rings that have helped form the foundation for much of contemporary research on the representation theory of Artinian rings and finite-dimensional algebras.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Указатель в процессе заполнения

ed2k: ed2k stats

Издание: 2-nd edition

Год издания: 1992

Количество страниц: 380

Добавлена в каталог: 20.11.2009

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
(M-) complement of N      75
Additive functor      177 180
Adjoint functor      247 253
Admissible sequence      349
Algebra      17
Algebra semigroup      359
Algebra, artin algebra      333
Algebra, Frobenius algebra      344
Algebra, path algebra of a quiver      360
Algebra, semigroup algebra      353
Algebra, split algebra      353
Algebra, split serial algebra      353
Algebra, symmetric algebra      345
Annihilator(s)      35 37 233 280 288
Annihilator(s), annihilator(s) of M in R      35
Annihilator(s), annihilator(s) with respect to $\bigotimes$      233
Annihilator(s), double annihilator(s) property      286
Annihilator(s), left annihilator(s) of X in R      37
Annihilator(s), maximum condition for annihilator      322
Annihilator(s), right annihilator(s) of A in M      37
Artin — Wedderburn theorem      154 309
Artinian, artinian module      127
Artinian, artinian ring      129 172
Artinian, simple artinian ring      153 160 218
Ascending chain condition      127
Automorphism      58
Axiom of Choice      2
Azumaya — Mority duality for artinian rings      §30 331
Azumaya's Theorem      144
Baer's criterion      205
Balanced, balanced bimodule      60
Balanced, balanced module      61
Balanced, R-balanced function      218
Bass's Theorem P      315
Biendomorphism      60 157
Bijection      2
Bilinear map      280
Bilinear map, non-degenerate bilinear map      327
Bilinear map, symmetric bilinear map      345
Bimodule      28 36 59
Bimodule, balanced bimodule      60
Bimodule, canonical bimodule      237
Bimodule, faithful bimodule      60
Boolean, boolean ring      24 133
Boolean, complete boolean ring      217
C-generated module      292 296
Cancellable      11
Capital      120
Cardinal numbers      5
Cartan matrix      358
Cartesian product      2 79
Cartesian product, cartesian product module      28 79
Cartesian product, projections for cartesian product      2 67 see
Category      6
Category, categories of left R-modules      55
Category, category equivalence      250 254
Category, category of finitely spanned modules      55
Category, concrete category      6
Category, duality of categories      269
Category, equivalent categories      250
Category, full category      7
Category, functor between categories      7
Category, hull category      277
Category, morphism in a category      6
Category, objects of a category      6
Category, product of categories      260
Category, subcategory of a category      7
Category, underlying category      7
Center of a ring      17
centralizer      24
Chain      3
Chain conditions      127 129
Chain conditions, ascending chain condition      127
Chain conditions, descending chain condition      127
Chain conditions, posets with chain conditions      132
Chain conditions, rings with chain conditions      129
Characteristic of a ring      24
Chinese remainder theorem      103
Co-artinian      217
Co-domain      6
Co-faithful      217
Co-generate      106 256
Co-generate, finitely co-generate      106 124
co-generator      107
Co-generator, co-generator module      210
Co-generator, co-generator ring      286
Co-generator, minimal co-generator      211
Co-noetherian      217
Co-semisimple      122 156 216
Coimage      43
Coindependent      94
Cokernel      43
Comaximal ideals      103
Complement direct summands      141
Composition, composition factors      135
Composition, composition length      136 137
Composition, composition series      134
Composition, Jordan — Hoelder Theorem      135
Composition, length of composition series      134 136
Concrete category      6
Concrete category, concrete category of groups      7
Concrete category, concrete category of real vector spaces      7
Concrete category, concrete category of sets      7
Coproduct      82 105
Countably generated module      295 297
Crawley — Jonsson — Warfield Theorem      298
Cyclic module      32 47
Decomposition, block decomposition      100 130
Decomposition, decomposition of injective modules      290
Decomposition, decomposition of rings      §7 98
Decomposition, direct decomposition      86
Decomposition, idempotents for a decomposition      88
Decomposition, indecomposable decomposition      140
Dedekind domain      215
Density Theorem (Jacobson — Chevalley)      159 160
Descending chain condition      127
Diagram      1
Diagram, commutative diagram      1
Diagram, Hasse diagram      39
Dimension Theorem      137
Direct decomposition      86
Direct product, coordinate injections for a direct product      80
Direct product, direct product homomorphism      79 80
Direct product, direct product of additive factors      230
Direct product, direct product of modules      79
Direct product, sub direct product      104
Direct product, universal mapping property of a direct product      80
Direct sum, direct sum (coproduct)      82 89
Direct sum, direct sum homomorphism      83
Direct sum, direct sum of additive functors      236
Direct sum, direct sum of modules      66 82 298
Direct sum, external direct sum      83
Direct sum, indecomposable module      71
Direct sum, internal direct sum      66 84
Direct sum, projections for a direct sum      69
Direct sum, ring direct sum      98 130
Direct summand      66
Direct summand, complementary direct summand      66
Divisible group      205
Divisible R-module      54
Division ring      11 161
Domain      6
Domain, Dedekind domain      215
Domain, principal ideal domain      114
Dual basis lemma      203
Duality, Azumaya — Morita Duality Theorem      331
Duality, duality of categories      269
Duality, Morita duality      §24 278
Endomorphism      20 58
Endomorphism, endomorphism of a module (convention)      58
Endomorphism, idempotent endomorphism      70
Endomorphism, local endomorphism ring      297
Endomorphism, rank of endomorphism      164
Endomorphism, ring of endomorphisms      20 213 236 297 322
Epimorphism      43
Epimorphism, epimorphism in a category      51 276
Epimorphism, natural epimorphism      43
Epimorphism, split epimorphism      66
Epimorphism, superfluous epimorphism      73
Equivalence(s), category equivalence(s)      250
Equivalence(s), equivalence map      140
Equivalence(s), inverse category equivalences      250
Equivalent, (Morita) equivalent rings      251
Equivalent, equivalent categories      250
Equivalent, equivalent composition series      135
Equivalent, equivalent decompositions      140
Equivalent, equivalent elements in a domain      215
Essential, essential monomorphism      73
Essential, essential submodule      72
Evaluation map      244 270
Exactness      49
Exactness, exactness test for $\bigotimes$      226
Exactness, exactness under category equivalence      254
Exactness, extension      49
Exactness, short exact sequence      49
Exactness, split exactness      67
Extension(s), equivalent extensions      53
Extension(s), extension(s) of K by N      49
Extension(s), transitive extension      103
External direct sum      83
Factor Theorem      45
Factor, factor module      33
Factor, factor ring      14
Faith — Walker characterization of QF rings      343
Faith — Walker Theorem      293
Faithful, *-faithful      217
Faithful, co-faithful      217
Faithful, completely faithful      233
Faithful, faithful functor      247
Faithful, faithful module      35 61 233
Field      11
Field, field of fractions      11
Finite module type      326
Fitting's lemma      138 139
Five lemma      50 52
Flat module      226
Flat Test Lemma      227
Free R-module      113 300
Function(s)      1
Function(s), bijective function      2
Function(s), bilinear function      280
Function(s), composite of functions      1
Function(s), function ring      16
Function(s), identity function      1
Function(s), inclusion functions      2
Function(s), injective function      1
Function(s), Kronecker delta function      2
Function(s), order preserving (reversing) function(s)      4
Function(s), product of functions      1
Function(s), restriction of a function      1
Function(s), support of a function      24
Function(s), surjective function      1
Functor(s)      7 §16 177 324
Functor(s), additive functor      177 180
Functor(s), adjoint functor(s)      247 253
Functor(s), contravariant functor      8
Functor(s), covariant functor      7
Functor(s), direct sum % product of functors      236
Functor(s), dual functor(s)      244
Functor(s), exact functor      183
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