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Dimensions of ring theory

Author: C Năstăsescu; F Van Oystaeyen
Publisher: Dordrecht, Holland ; Boston : D. Reidel Pub. Co. ; Norwell, MA : Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, ©1987.
Series: Mathematics and its applications (D. Reidel Publishing Company)
Edition/Format:   Print book : EnglishView all editions and formats
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Further, the kind and level of s9phistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical  Read more...

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Additional Physical Format: Online version:
Năstăsescu, C. (Constantin), 1943-
Dimensions of ring theory.
Dordrecht, Holland ; Boston : D. Reidel Pub. Co. ; Norwell, MA : Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, ©1987
(OCoLC)752444002
Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: C Năstăsescu; F Van Oystaeyen
ISBN: 902772461X 9789027724618 0691077509 9780691077505 0691022747 9780691022741
OCLC Number: 15224434
Description: xi, 360 pages ; 25 cm.
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.- 4.2.1. The Ring of Endomorphisms of a Projective Finitely Generated Module.- 4.2.2. Normalizing Extensions.- 4.2.3. Rings Strongly Graded by a Finite Group.- 4.2.4. The Ring of Invariants.- 4.2.5. Graded Rings of Type ?.- 4.2.6. Filtered Rings and Modules.- 4.2.7. Ore and Skew-Laurent Extensions.- 4.2.8. Affine P.I. Algebras.(Addendum).- 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.- 10.3.1. Rings of Differential Operators and Weyl Algebras.- 10.3.2. Remarks on Enveloping Algebras of Lie Algebras.(Addendum)..- 10.3.3. P.I.Algebras.(Addendum).- 10.4. Exercises.- Bibliographical Comments to Chapter 10.- References.
Series Title: Mathematics and its applications (D. Reidel Publishing Company)
Responsibility: by Constantin Năstăsescu and Freddy Van Oystaeyen.
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