Finite Commutative Rings and Their Applications

by Bini, Gilberto; Flamini, Flaminio

• ISBN13: 9781402070396
• ISBN10: 140207039X
• Format: Hardcover
• Copyright: 2002-04-01
• Publisher: Kluwer Academic Pub

Summary

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 The section on Quasi-Galois rings presents new and unpublished results as well.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

Table of Contents

Foreword	vii
Preface	ix
Fundamental Notions in Ring Theory	1	(28)
Basic Definitions	1	(2)
Prime and Maximal Ideals	3	(6)
Euclidean Domains, P.I.D.'s and U.F.D.'s	9	(11)
Factorization in Zpn [x]	20	(9)
Finite Field Structure	29	(10)
Basic Properties	29	(2)
Characterization of Finite Fields	31	(3)
Galois Field Automorphisms	34	(5)
Finite Commutative Rings. Regular Polynomials	39	(22)
Finite Commutative Ring Structure	39	(8)
Regular Polynomials in the Ring R[x]	47	(7)
R-algebra Automorphisms of R[x]	54	(2)
Factorization in R[x]	56	(5)
Separable Extensions of Finite Fields and a Finite Rings	61	(10)
Separable Field Extensions	61	(4)
Extensions of Rings	65	(2)
Separable extensions of finite commutative local rings	67	(4)
Galois Theory for Local Rings	71	(10)
Basic Facts	71	(4)
Examples. Splitting Rings	75	(6)
Galois and Quasi-Galois Rings: Structure and Properties	81	(40)
Classical Constructions	82	(10)
Galois Ring Properties	92	(13)
Structure Theorems for finite commutative local rings	105	(2)
Another class of finite commutative local rings: Quasi-Galois Rings	107	(14)
Basic Notions on Codes Over Finite Fields	121	(24)
Basic properties	121	(1)
Some families of q-ary codes	122	(8)
Linear Codes	122	(1)
Hamming codes	123	(1)
Cyclic codes	124	(5)
Reed-Muller codes	129	(1)
Duality between codes	130	(4)
Some families of nonlinear q-ary codes	134	(11)
Binary Kerdock codes	135	(1)
Kerdock sets	135	(3)
Properties of binary Kerdock codes	138	(2)
Classical Preparata codes	140	(1)
Basic properties	141	(1)
Preparata codes and Hamming codes	142	(3)
Basic Notions on Codes Over Galois Rings	145	(22)
Basic properties	145	(7)
Linear codes over Zpn	146	(1)
Reed-Muller codes over Zpn	147	(1)
Cyclic codes over Zpn	148	(3)
Hamming codes over Zpn	151	(1)
Linear quaternary codes	152	(6)
Kerdock and Preparata codes revisited	158	(9)
Bibliography	167	(6)
Index	173