Author | Nวstวsescu, Constantin. author |
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Title | Dimensions of Ring Theory [electronic resource] / by Constantin Nวstวsescu, Freddy van Oystaeyen |
Imprint | Dordrecht : Springer Netherlands, 1987 |
Connect to | http://dx.doi.org/10.1007/978-94-009-3835-9 |
Descript | XI, 360 p. online resource |
1. Finiteness Conditions for Lattices -- 1.1. Lattices -- 1.2. Noetherian and Artinian Lattices -- 1.3. Lattices of Finite Length -- 1.4. Irreducible Elements in a Lattice -- 1.5. Goldie Dimension of a Modular Lattice -- 1.6. Goldie Dimension and Chain Conditions for Modular Lattices with Finite Group Actions -- 1.7. Complements and Pseudo-Complements -- 1.8. Semiatomic Lattices and Compactly Generated Lattices -- 1.9. Semiartinian Lattices -- 1.10. Indecomposable Elements in a Lattice -- 1.11. Exercises -- Bibliographical Comments to Chapter 1 -- 2. Finiteness Conditions for Modules -- 2.1. Modules -- 2.2. The Lattice of Submodules of a Module -- 2.3. Noetherian and Artinian Modules -- 2.4. Modules of Finite Length -- 2.5. Semisimple Modules -- 2.6. Semisimple and Simple Artinian Rings -- 2.7. The Jacobson Radical and the Prime Radical of a Ring -- 2.8. Rings of Fractions. Goldieโs Theorems -- 2.9. Artinian Modules which are Noetherian -- 2.10. Projective and Infective Modules -- 2.11. Tensor Product and Flat Modules -- 2.12. Normalizing Extensions of a Ring -- 2.13. Graded Rings and Modules -- 2.14. Graded Rings and Modules of Type ?. Internal Homogenisation -- 2.15. Noetherian Modules over Graded Rings of Type ?. Applications -- 2.16. Strongly Graded Rings and Clifford Systems for Finite Groups -- 2.17. Invariants of a Finite Group Action -- 2.18. Exercises -- Bibliographical Comments to Chapter 2 -- 3. Krull Dimension and Gabriel Dimension of an Ordered Set -- 3.1. Definitions and Basic Properties -- 3.2. The Krull Dimension of a Modular Lattice -- 3.3. Critical Composition Series of a Lattice -- 3.4. The Gabriel Dimension of a Modular Lattice -- 3.5. Comparison of Krull and Gabriel Dimension -- 3.6. Exercises -- Bibliographical Comments to Chapter 3 -- 4. Krull Dimension and Gabriel Dimension of Rings and Modules -- 4.1. Definitions and Generalities -- 4.2. Krull and Gabriel Dimension of Some Special Classes of Rings and Modules -- 4.3. Exercises -- Bibliographical Comments to Chapter 4 -- 5. Rings with Krull Dimension -- 5.1. Nil Ideals -- 5.2. Semiprime Rings with Krull Dimension -- 5.3. Classical Krull Dimension of a Ring -- 5.4. Associated prime Ideals -- 5.5. Fully Left Bounded Rings with Krull Dimension -- 5.6. Examples of Noetherian Rings of Arbitrary Krull Dimension -- 5.7. Exercises -- Bibliographical Comments to Chapter 5 -- 6. Krull Dimension of Noetherian Rings. The Principal Ideal Theorem -- 6.1. Fully Left Bounded Left Noetherian Rings -- 6.2. The Reduced Rank of a Module -- 6.3. Noetherian Rings Satisfying Condition H -- 6.4. Fully Bounded Noetherian Rings -- 6.5. Krull Dimension and Invertible Ideals in a Noetherian Ring -- 6.6. The Principal Ideal Theorem -- 6.7. Exercises -- Bibliographical Comments to Chapter 6 -- 7. Relative Krull and Gabriel Dimensions -- 7.1. Additive Topologies and Torsion Theories -- 7.2. The Lattices CF (M) and CHg -- 7.3. Relative Krull Dimension -- 7.4. Relative Krull Dimension Applied to the Principal Ideal Theorem -- 7.5. Relative Gabriel Dimension -- 7.6. Relative Krull and Gabriel Dimensions of Graded Rings -- 7.7. Exercises -- Bibliographical Comments to Chapter 7 -- 8. Homological Dimensions -- 8.1. The Projective Dimension of a Module -- 8.2. Homological Dimension of Polynomial Rings and Rings of Formal Power Series -- 8.3. Injective Dimension of a Module -- 8.4. The Flat Dimension of a Module -- 8.5. The Artin-Rees Property and Homological Dimensions -- 8.6. Regular Local Rings -- 8.7. Exercises -- Bibliographical Comments to Chapter 8 -- 9. Rings of Finite Global Dimension -- 9.1. The Zariski Topology -- 9.2. The Local Study of Homological Dimension -- 9.3. Rings Integral over their Centres -- 9.4. Commutative Rings of Finite Global Dimension -- 9.5. Exercises -- Bibliographical Comments to Chapter 9 -- 10. The Gelfand-Kirillov Dimension -- 10.1. Definitions and Basic Properties -- 10.2. GK-dimension of Filtered and Graded Algebras -- 10.3. Applications to Special Classes of Rings -- 10.4. Exercises -- Bibliographical Comments to Chapter 10 -- References