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Dimensions of ring theory / by Constantin Nǎstǎsescu, Freddy Oystaeyen.

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G.K. Chesterton. The Scandal of Father 'The Hermit Gad in Crane Feathers...

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Bibliographic Details
Online Access: Full Text (via Springer)
Main Authors: Năstăsescu, C. (Constantin), 1943- (Author), Oystaeyen, F. Van, 1947- (Author)
Format: eBook
Language:English
Published: Dordrecht : D. Reidel Publishing Company, [1987]
Series:Mathematics and its applications (D. Reidel Publishing Company)
Subjects:
Rings (Algebra)
Dimension theory (Algebra)
Algebra.
Mathematics.
Table of Contents:
  • 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. Goldie's 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.

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