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.
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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