Finite Commutative Rings and Their Applications

ISBN-13: 9781402070396 / Angielski / Twarda / 2002 / 176 str.

Gilberto Bini; Flaminio Flamini

Opis

Foreword by Dieter Jungnickel
Finite Commutative Rings and their Applications answers a need for an introductory reference in finite commutative ring theory as applied to information and communication theory. This book will be of interest to both professional and academic researchers in the fields of communication and coding theory. The book is a concrete and self-contained introduction to finite commutative local rings, focusing in particular on Galois and Quasi-Galois rings. The reader is provided with an active and concrete approach to the study of the purely algebraic structure and properties of finite commutative rings (in particular, Galois rings) as well as to their applications to coding theory. Finite Commutative Rings and their Applications is the first to address both theoretical and practical aspects of finite ring theory. The authors provide a practical approach to finite rings through explanatory examples, thereby avoiding an abstract presentation of the subject. The section on Quasi-Galois rings presents new and unpublished results as well. The authors then introduce some applications of finite rings, in particular Galois rings, to coding theory, using a solid algebraic and geometric theoretical background. This text is suitable for courses in commutative algebra, finite commutative algebra, and coding theory. It is also suitable as a supplementary text for courses in discrete mathematics, finite fields, finite rings, etc.

Spis treści

Foreword. Preface. 1. Fundamental Notions in Ring Theory. 2. Finite Field Structure. 3. Finite Commutative Rings. Regular Polynomials. 4. Separable Extensions of Finite Fields and Finite Rings. 5. Galois Theory for Local Rings. 6. Galois and Quasi-Galois Rings: Structure and Properties. 7. Basic Notions of Codes Over Finite Fields. 8. Basic Notions on Codes over Galois Rings. Bibliography. Index.