Author | Stenstrรถm, Bo. author |
---|---|
Title | Rings of Quotients [electronic resource] : An Introduction to Methods of Ring Theory / by Bo Stenstrรถm |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1975 |
Connect to | http://dx.doi.org/10.1007/978-3-642-66066-5 |
Descript | 309 p. online resource |
Notations and Conventions -- I. Modules -- ยง 1. Basic Definitions -- ยง 2. Sums and Products of Modules -- ยง 3. Finitely Generated Modules and Noetherian Modules -- ยง 4. Categories and Functors -- ยง 5. Exactness of Functors between Module Categories -- ยง 6. Projective and Injective Modules -- ยง 7. Semi-Simple Modules and Rings -- ยง 8. Tensor Products -- ยง 9. Bimodules -- ยง 10. Flat Modules -- ยง 11. Pure Submodules -- ยง 12. Regular Rings -- ยง 13. Coherent Rings -- Exercises -- II. Rings of Fractions -- ยง 1. The Ring of Fractions -- ยง 2. Orders in a Semi-Simple Ring -- ยง 3. Modules of Fractions -- ยง 4. Invertible Ideals and Hereditary Orders -- Exercises -- III. Modular Lattices -- ยง 1. Lattices -- ยง 2. Modularity -- ยง 3. Lattices with Chain Condition -- ยง 4. Distributive Lattices -- ยง 5. Continuous Lattices -- ยง 6. Pseudo-Complemented Lattices -- ยง 7. Closure Operators -- ยง 8. Galois Connections -- Exercises -- IV. Abelian Categories -- ยง 1. Equivalence of Categories -- ยง 2. Kernels and Cokernels -- ยง 3. Products and Coproducts -- ยง 4. Abelian Categories -- ยง 5. Pullbacks and Pushouts -- ยง 6. Generators and Cogenerators -- ยง 7. Functor Categories -- ยง 8. Limits and Colimits -- ยง 9. Adjoint Functors -- ยง 10. Morita Equivalence -- Exercises -- V. Grothendieck Categories -- ยง 1. Exactness of Direct Limits -- ยง 2. Injective Envelopes -- ยง 3. Finitely Generated Objects -- ยง 4. Locally Noetherian Categories -- ยง 5. The Krull-Remak-Schmidt-Azumaya Theorem -- ยง 6. Spectral Categories -- ยง 7. The Spectral Category of a Grothendieck Category -- Exercises -- VI. Torsion Theory -- ยง 1. Preradicals -- ยง 2. Torsion Theories -- ยง 3. Hereditary Torsion Theories -- ยง 4. Linear Topologies -- ยง 5. Gabriel Topologies -- ยง 6. Examples of Gabriel Topologies -- ยง 7. Stable Torsion -- ยง 8. TTF-Classes -- ยง 9. ?-Divisible Modules -- Exercises -- VII. Hereditary Torsion Theories for Noetherian Rings -- ยง 1. Associated Prime Ideals -- ยง 2. Fully Bounded Noetherian Rings -- ยง 3. Topologies for a Fully Bounded Noetherian Ring -- ยง 4. Artin-Rees Modules and Stable Topologies -- ยง 5. Auxiliary Results on Commutative Localization -- ยง 6. The Topologies ?Mn for a Commutative Noetherian Ring -- Exercises -- VIII. Simple Torsion Theories -- ยง 1. The Jacobson Radical and Artinian Rings -- ยง 2. Semi-Artinian Modules and Rings -- ยง 3. Simple Torsion Theories -- ยง 4. Semi-Perfect Rings -- ยง 5. Perfect Rings -- ยง 6. Hereditary Torsion Theories for a Perfect Ring -- Exercises -- IX. Rings and Modules of Quotients -- ยง 1. Construction of Modules of Quotients -- ยง 2. ?-Injective Envelopes -- ยง 3. The Ring of Quotients is a Bicommutator -- ยง 4. The Lattice of Saturated Submodules -- ยง 5. ?-Invertible Ideals -- Exercises -- X. The Category of Modules of Quotients -- ยง 1. Giraud Subcategories -- ยง 2. Gabriel Topologies and Giraud Subcategories -- ยง 3. Rings of Quotients of Morita Equivalent Rings -- ยง 4. Representation of Grothendieck Categories -- Exercises -- XI. Perfect Localizations -- ยง 1. Epimorphisms of Rings -- ยง 2. Flat Epimorphisms of Rings -- ยง 3. Perfect Gabriel Topologies -- ยง 4. The Maximal Flat Epimorphic Ring of Quotients -- ยง 5. The Maximal Ring of Quotients as a Perfect Localization -- ยง 6. 1-Topologies and Rings of Fractions -- ยง 7. Finite Localizations -- ยง 8. The Endomorphism Ring of a Finitely Generated Projective Module -- Exercises -- XII. The Maximal Ring of Quotients of a Non-Singular Ring -- ยง 1. Standard Representation of Spectral Categories -- ยง 2. The Maximal Ring of Quotients -- ยง 3. The Maximal Ring of Quotients of a Boolean Ring -- ยง 4. The Lattice of Essentially Closed Ideals -- ยง 5. The Maximal Ring of Quotients of a Reduced Ring -- ยง 6. Flatness of the Maximal Ring of Quotients -- ยง 7. Imbedding Non-Singular Modules in Free Modules -- Exercises -- XIII. Finiteness Conditions on Mod-(A, ?) -- ยง 1. Compact Lattices of Saturated Submodules -- ยง 2. ?-Noetherian Rings -- ยง 3. The Goldie Topology -- ยง 4. P-Noetherian Rings -- ยง 5. The Canonical Topology of a Krull Domain -- Exercises -- XIV. Self-Injective Rings -- ยง 1. The Endomorphism Ring of an Injective Module -- ยง 2. Self-Injective Rings and Annihilator Conditions -- ยง 3. Quasi-Frobenius Rings -- ยง 4. Self-Injective Maximal Rings of Quotients -- Exercises -- XV. Classical Rings of Quotients -- ยง 1. Semi-Prime Ideals and the Prime Radical -- ยง 2. Rings of Quotients of Quotient Rings -- ยง 3. Orders in a Semi-Primary Ring -- ยง 4. Noetherian Orders in an Artinian Ring -- ยง 5. Orders in Self-Injective Rings -- ยง 6. Rings of Quotients of Group Algebras -- Exercises