Ring constructions and applications.

*(English)*Zbl 0999.16036
Series in Algebra. 9. Singapore: World Scientific. xii, 205 p. (2002).

This is a monograph that may be of use for any mathematician interested in (associative) rings and their applications.

From the preface: “This book is devoted to various ring constructions, their properties, and examples of applications. It would be really valuable to find a single viewpoint which makes it possible to achieve good understanding of many notions in a unified framework. The emphasis is on the fairly new concept of a groupoid-graded ring embracing and unifying a variety of constructions…. Groupoid-graded rings include as special cases many other ring constructions: polynomial and skew polynomial rings, Ore extensions, direct and semidirect products, matrix and structural matrix rings, Rees matrix rings, Morita contexts and generalized matrix rings, group and semigroup rings, skew group and semigroup rings, twisted group and semigroup rings, monomial rings, smash products, crossed products, group graded rings, monoid graded rings, path algebras, edge algebras, incidence rings, etc. Therefore groupoid-graded rings can be applied to the study of other less general constructions.”

Contents. Preface. Chapter 1 Preliminaries: 1.1 Groupoids, 1.2 Groups, 1.3 Semigroups, 1.4 Rings. Chapter 2 Graded rings: 2.1 Groupoid-graded rings, 2.2 Semigroup-graded rings, superalgebras. Chapter 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. Chapter 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. Chapter 5 Groups of units. Chapter 6 Finiteness conditions: 6.1 Groupoid-graded rings 6.2 Structural approach of Jespers and Okniński, 6.3 Finiteness conditions and homogeneous components, 6.4 Classical Krull dimension. Chapter 7 PI-rings and varieties. Chapter 8 Gradings of matrix rings; 8.1 Full and upper triangular matrix ring, 8.2 Gradings by two-element semigroups, 8.3 Structural matrix superalgebras, Chapter 9: 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. Chapter 10 Open problems. Appendix A Glossary of notation. Bibliography. Index.

From the preface: “This book is devoted to various ring constructions, their properties, and examples of applications. It would be really valuable to find a single viewpoint which makes it possible to achieve good understanding of many notions in a unified framework. The emphasis is on the fairly new concept of a groupoid-graded ring embracing and unifying a variety of constructions…. Groupoid-graded rings include as special cases many other ring constructions: polynomial and skew polynomial rings, Ore extensions, direct and semidirect products, matrix and structural matrix rings, Rees matrix rings, Morita contexts and generalized matrix rings, group and semigroup rings, skew group and semigroup rings, twisted group and semigroup rings, monomial rings, smash products, crossed products, group graded rings, monoid graded rings, path algebras, edge algebras, incidence rings, etc. Therefore groupoid-graded rings can be applied to the study of other less general constructions.”

Contents. Preface. Chapter 1 Preliminaries: 1.1 Groupoids, 1.2 Groups, 1.3 Semigroups, 1.4 Rings. Chapter 2 Graded rings: 2.1 Groupoid-graded rings, 2.2 Semigroup-graded rings, superalgebras. Chapter 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. Chapter 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. Chapter 5 Groups of units. Chapter 6 Finiteness conditions: 6.1 Groupoid-graded rings 6.2 Structural approach of Jespers and Okniński, 6.3 Finiteness conditions and homogeneous components, 6.4 Classical Krull dimension. Chapter 7 PI-rings and varieties. Chapter 8 Gradings of matrix rings; 8.1 Full and upper triangular matrix ring, 8.2 Gradings by two-element semigroups, 8.3 Structural matrix superalgebras, Chapter 9: 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. Chapter 10 Open problems. Appendix A Glossary of notation. Bibliography. Index.

Reviewer: Josif S.Ponizovskii (St.Peterburg)

##### MSC:

16W50 | Graded rings and modules (associative rings and algebras) |

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

16S34 | Group rings |

16S35 | Twisted and skew group rings, crossed products |

16S36 | Ordinary and skew polynomial rings and semigroup rings |

16S40 | Smash products of general Hopf actions |

20M25 | Semigroup rings, multiplicative semigroups of rings |

16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |