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.
(M-) complement of N75 Additive functor177180 Adjoint functor247253 Admissible sequence349 Algebra17 Algebra semigroup359 Algebra, artin algebra333 Algebra, Frobenius algebra344 Algebra, path algebra of a quiver360 Algebra, semigroup algebra353 Algebra, split algebra353 Algebra, split serial algebra353 Algebra, symmetric algebra345 Annihilator(s)3537233280288 Annihilator(s), annihilator(s) of M in R35 Annihilator(s), annihilator(s) with respect to 233 Annihilator(s), double annihilator(s) property286 Annihilator(s), left annihilator(s) of X in R37 Annihilator(s), maximum condition for annihilator322 Annihilator(s), right annihilator(s) of A in M37 Artin — Wedderburn theorem154309 Artinian, artinian module127 Artinian, artinian ring129172 Artinian, simple artinian ring153160218 Ascending chain condition127 Automorphism58 Axiom of Choice2 Azumaya — Mority duality for artinian rings§30331 Azumaya's Theorem144 Baer's criterion205 Balanced, balanced bimodule60 Balanced, balanced module61 Balanced, R-balanced function218 Bass's Theorem P315 Biendomorphism60157 Bijection2 Bilinear map280 Bilinear map, non-degenerate bilinear map327 Bilinear map, symmetric bilinear map345 Bimodule283659 Bimodule, balanced bimodule60 Bimodule, canonical bimodule237 Bimodule, faithful bimodule60 Boolean, boolean ring24133 Boolean, complete boolean ring217 C-generated module292296 Cancellable11 Capital120 Cardinal numbers5 Cartan matrix358 Cartesian product279 Cartesian product, cartesian product module2879 Cartesian product, projections for cartesian product267see Category6 Category, categories of left R-modules55 Category, category equivalence250254 Category, category of finitely spanned modules55 Category, concrete category6 Category, duality of categories269 Category, equivalent categories250 Category, full category7 Category, functor between categories7 Category, hull category277 Category, morphism in a category6 Category, objects of a category6 Category, product of categories260 Category, subcategory of a category7 Category, underlying category7 Center of a ring17 centralizer24 Chain3 Chain conditions127129 Chain conditions, ascending chain condition127 Chain conditions, descending chain condition127 Chain conditions, posets with chain conditions132 Chain conditions, rings with chain conditions129 Characteristic of a ring24 Chinese remainder theorem103 Co-artinian217 Co-domain6 Co-faithful217 Co-generate106256 Co-generate, finitely co-generate106124 co-generator107 Co-generator, co-generator module210 Co-generator, co-generator ring286 Co-generator, minimal co-generator211 Co-noetherian217 Co-semisimple122156216 Coimage43 Coindependent94 Cokernel43 Comaximal ideals103 Complement direct summands141 Composition, composition factors135 Composition, composition length136137 Composition, composition series134 Composition, Jordan — Hoelder Theorem135 Composition, length of composition series134136 Concrete category6 Concrete category, concrete category of groups7 Concrete category, concrete category of real vector spaces7 Concrete category, concrete category of sets7 Coproduct82105 Countably generated module295297 Crawley — Jonsson — Warfield Theorem298 Cyclic module3247 Decomposition, block decomposition100130 Decomposition, decomposition of injective modules290 Decomposition, decomposition of rings§798
Decomposition, direct decomposition86 Decomposition, idempotents for a decomposition88 Decomposition, indecomposable decomposition140 Dedekind domain215 Density Theorem (Jacobson — Chevalley)159160 Descending chain condition127 Diagram1 Diagram, commutative diagram1 Diagram, Hasse diagram39 Dimension Theorem137 Direct decomposition86 Direct product, coordinate injections for a direct product80 Direct product, direct product homomorphism7980 Direct product, direct product of additive factors230 Direct product, direct product of modules79 Direct product, sub direct product104 Direct product, universal mapping property of a direct product80 Direct sum, direct sum (coproduct)8289 Direct sum, direct sum homomorphism83 Direct sum, direct sum of additive functors236 Direct sum, direct sum of modules6682298 Direct sum, external direct sum83 Direct sum, indecomposable module71 Direct sum, internal direct sum6684 Direct sum, projections for a direct sum69 Direct sum, ring direct sum98130 Direct summand66 Direct summand, complementary direct summand66 Divisible group205 Divisible R-module54 Division ring11161 Domain6 Domain, Dedekind domain215 Domain, principal ideal domain114 Dual basis lemma203 Duality, Azumaya — Morita Duality Theorem331 Duality, duality of categories269 Duality, Morita duality§24278 Endomorphism2058 Endomorphism, endomorphism of a module (convention)58 Endomorphism, idempotent endomorphism70 Endomorphism, local endomorphism ring297 Endomorphism, rank of endomorphism164 Endomorphism, ring of endomorphisms20213236297322 Epimorphism43 Epimorphism, epimorphism in a category51276 Epimorphism, natural epimorphism43 Epimorphism, split epimorphism66 Epimorphism, superfluous epimorphism73 Equivalence(s), category equivalence(s)250 Equivalence(s), equivalence map140 Equivalence(s), inverse category equivalences250 Equivalent, (Morita) equivalent rings251 Equivalent, equivalent categories250 Equivalent, equivalent composition series135 Equivalent, equivalent decompositions140 Equivalent, equivalent elements in a domain215 Essential, essential monomorphism73 Essential, essential submodule72 Evaluation map244270 Exactness49 Exactness, exactness test for 226 Exactness, exactness under category equivalence254 Exactness, extension49 Exactness, short exact sequence49 Exactness, split exactness67 Extension(s), equivalent extensions53 Extension(s), extension(s) of K by N49 Extension(s), transitive extension103 External direct sum83 Factor Theorem45 Factor, factor module33 Factor, factor ring14 Faith — Walker characterization of QF rings343 Faith — Walker Theorem293 Faithful, *-faithful217 Faithful, co-faithful217 Faithful, completely faithful233 Faithful, faithful functor247 Faithful, faithful module3561233 Field11 Field, field of fractions11 Finite module type326 Fitting's lemma138139 Five lemma5052 Flat module226 Flat Test Lemma227 Free R-module113300 Function(s)1 Function(s), bijective function2 Function(s), bilinear function280 Function(s), composite of functions1 Function(s), function ring16 Function(s), identity function1 Function(s), inclusion functions2 Function(s), injective function1 Function(s), Kronecker delta function2 Function(s), order preserving (reversing) function(s)4 Function(s), product of functions1 Function(s), restriction of a function1 Function(s), support of a function24 Function(s), surjective function1 Functor(s)7§16177324 Functor(s), additive functor177180 Functor(s), adjoint functor(s)247253 Functor(s), contravariant functor8 Functor(s), covariant functor7 Functor(s), direct sum % product of functors236 Functor(s), dual functor(s)244 Functor(s), exact functor183
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