Finite Commutative Rings and Their Applications

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 The section on Quasi-Galois rings presents new and unpublished results as well.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.

1 Fundamental Notions In Ring Theory.- 1.1 Basic Definitions.- 1.2 Prime and Maximal Ideals.- 1.3 Euclidean Domains, P.I.D.’s and U.F.D.’s.- 1.4 Factorization in $$ {Z_p}n $$[x].- 2 Finite Field Structure.- 2.1 Basic Properties.- 2.2 Characterization of Finite Fields.- 2.3 Galois Field Automorphisms.- 3 Finite Commutative Rings. Regular Polynomials.- 3.1 Finite Commutative Ring Structure.- 3.2 Regular Polynomials in the Ring R[x].- 3.3 R-algebra Automorphisms of R[x].- 3.4 Factorization in R[x].- 4 Separable Extensions of Finite Fields and Finite Rings.- 4.1 Separable Field Extensions.- 4.2 Extensions of Rings.- 4.3 Separable extensions of finite commutative local rings.- 5 Galois Theory for Local Rings.- 5.1 Basic Facts.- 5.2 Examples. Splitting Rings.- 6 Galois and Quasi-Galois Rings: Structure and Properties.- 6.1 Classical Constructions.- 6.2 Galois Ring Properties.- 6.3 Structure Theorems for finite commutative local rings.- 6.4 Another class of finite commutative local rings: Quasi-Galois Rings.- 7 Basic Notions on Codes over Finite Fields.- 7.1 Basic properties.- 7.2 Some families of q-ary codes.- 7.3 Duality between codes.- 7.4 Some families of nonlinear q-ary codes.- 8 Basic Notions on Codes over Galois Rings.- 8.1 Basic properties.- 8.2 Linear quaternary codes.- 8.3 Kerdock and Preparata codes revisited.