Ring constructions and applications / Andrei V. Kelarev.

By: Kelarev, A. V. (Andrei V.)
Material type: TextTextSeries: Series in algebra: v. 9.Publisher: River Edge, N.J. : World Scientific, ©2002Description: 1 online resource (xi, 205 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9789812799722; 9812799729; 1281948047; 9781281948045; 9786611948047; 661194804XSubject(s): Rings (Algebra) | Anneaux (Algèbre) | MATHEMATICS -- Algebra -- Intermediate | Rings (Algebra) | ANÉIS E ÁLGEBRAS ASSOCIATIVOS | MÓDULOS (ÁLGEBRA)Genre/Form: Electronic books. | Electronic books. Additional physical formats: Print version:: Ring constructions and applications.DDC classification: 512/.4 LOC classification: QA247 | .K45 2002ebOnline resources: Click here to access online
Contents:
Ch. 1. Preliminaries. 1.1. Groupoids. 1.2. Groups. 1.3. Semigroups. 1.4. Rings -- ch. 2. Graded rings. 2.1 Groupoid-graded rings. 2.2. Semigroup-graded rings. 2.3. Group-graded rings. 2.4. Superalgebras -- ch. 3. Examples of ring constructions. 3.1. Direct, subdirect and semidirect products. 3.2. Group and semigroup rings, monomial rings. 3.3. Crossed products. 3.4. Polynomial and skew polynomial rings. 3.5. Skew group and semigroup rings. 3.6. Twisted group and semigroup rings. 3.7. Power and skew power series rings. 3.8. Edge and path algebras. 3.9. Matrix rings and generalized matrix rings. 3.10. Triangular matrix representations. 3.11. Morita contexts. 3.12. Rees matrix rings. 3.13. Smash products. 3.14. Structural matrix rings. 3.15. Incidence algebras -- ch. 4. The Jacobson radical. 4.1. The Jacobson radical of groupoid-graded rings. 4.2. Descriptions of the Jacobson radical. 4.3. Semisimple semigroup-graded rings. 4.4. Homogeneous radicals. 4.5. Radicals and homogeneous components. 4.6. Nilness and nilpotency -- ch. 5. Groups of units -- ch. 6. Finiteness conditions. 6.1. Groupoid-graded rings. 6.2. Structural approach of Jespers and Okninski. 6.3. Finiteness conditions and homogeneous components. 6.4. Classical Krull dimension and Gabriel dimension -- ch. 7. Pi-rings and varieties -- ch. 8. Gradings of matrix rings. 8.1. Full and upper triangular matrix rings. 8.2. Gradings by two-element semigroups. 8.3. Structural matrix superalgebras -- ch. 9. Examples of applications. 9.1. Codes as ideals in group rings. 9.2. Codes as ideals in matrix rings. 9.3. Color Lie superalgebras. 9.4. Combinatorial applications. 9.5. Applications in logic -- ch. 10. Open problems.
Summary: This book contains the definitions of several ring constructions used in various applications. The concept of a groupoid-graded ring includes many of these constructions as special cases and makes it possible to unify the exposition. Recent research results on groupoid-graded rings and more specialized constructions are presented. In addition, there is a chapter containing open problems currently considered in the literature. Ring Constructions and Applications can serve as an excellent introduction for graduate students to many ring constructions as well as to essential basic concepts of group, semigroup and ring theories used in proofs.
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Includes bibliographical references (pages 157-200) and index.

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Ch. 1. Preliminaries. 1.1. Groupoids. 1.2. Groups. 1.3. Semigroups. 1.4. Rings -- ch. 2. Graded rings. 2.1 Groupoid-graded rings. 2.2. Semigroup-graded rings. 2.3. Group-graded rings. 2.4. Superalgebras -- ch. 3. Examples of ring constructions. 3.1. Direct, subdirect and semidirect products. 3.2. Group and semigroup rings, monomial rings. 3.3. Crossed products. 3.4. Polynomial and skew polynomial rings. 3.5. Skew group and semigroup rings. 3.6. Twisted group and semigroup rings. 3.7. Power and skew power series rings. 3.8. Edge and path algebras. 3.9. Matrix rings and generalized matrix rings. 3.10. Triangular matrix representations. 3.11. Morita contexts. 3.12. Rees matrix rings. 3.13. Smash products. 3.14. Structural matrix rings. 3.15. Incidence algebras -- ch. 4. The Jacobson radical. 4.1. The Jacobson radical of groupoid-graded rings. 4.2. Descriptions of the Jacobson radical. 4.3. Semisimple semigroup-graded rings. 4.4. Homogeneous radicals. 4.5. Radicals and homogeneous components. 4.6. Nilness and nilpotency -- ch. 5. Groups of units -- ch. 6. Finiteness conditions. 6.1. Groupoid-graded rings. 6.2. Structural approach of Jespers and Okninski. 6.3. Finiteness conditions and homogeneous components. 6.4. Classical Krull dimension and Gabriel dimension -- ch. 7. Pi-rings and varieties -- ch. 8. Gradings of matrix rings. 8.1. Full and upper triangular matrix rings. 8.2. Gradings by two-element semigroups. 8.3. Structural matrix superalgebras -- ch. 9. Examples of applications. 9.1. Codes as ideals in group rings. 9.2. Codes as ideals in matrix rings. 9.3. Color Lie superalgebras. 9.4. Combinatorial applications. 9.5. Applications in logic -- ch. 10. Open problems.

This book contains the definitions of several ring constructions used in various applications. The concept of a groupoid-graded ring includes many of these constructions as special cases and makes it possible to unify the exposition. Recent research results on groupoid-graded rings and more specialized constructions are presented. In addition, there is a chapter containing open problems currently considered in the literature. Ring Constructions and Applications can serve as an excellent introduction for graduate students to many ring constructions as well as to essential basic concepts of group, semigroup and ring theories used in proofs.

English.

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