Finite commutative rings and their applications is the first to address both theoretical and practical aspects of. Classification of finite commutative rings mathoverflow. Algebraic complexities and algebraic curves over finite fields. All i know is a result that every finite commutative ring is a direct product of local commutative rings this is correct, right. Coding theory has its origins in the engineering problem of effective electronic communication where the alphabet is. Many areas of mathematics are used in coding theory, and we focus on the interplay between algebra. Finite commutative rings 1 introduction let s be a nonempty commutative semigroup, endowed with a binary associative operation let es be the set of idempotents of s, where e. Geometric langlands duality and representations of algebraic. Codes over chain rings are of special interest theoretically and practically since most known good codes are from chain rings. Algebraic coding theory revised 1984 edition by elwyn r. This text is intended for a one or twosemester undergraduate course in abstract algebra. Introduction to algebraic coding theory with gap fall 2006 sarah spence adams.
Quasigalois rings 107 7 basic notions on codes over finite fields 121. We define alternant codes over a commutative ring r and a corresponding key equation. Taylorthe arithmetic theory of local galois gauss sums for tame characters. Pdf commutative ring theory and applications download full. Geometric langlands duality and representations of. Lamprechtcalculation of general gauss sums and quadratic gauss sums in finite rings j. Cylinders, homotopies, and approximation theorems 121 3. This text is suitable for courses in commutative algebra, finite commutative algebra, and coding theory. Algebraic coding theory over finite commutative rings. Mathematics math ory, this volume is dedicated to codes constructed using algebraic geometry. The remainder of this chapter is structured as follows. Among the four axioms above, the first two are elementary consequences of the definition, whereas the remaining two hinge on important facts in commutative algebra, the goingup theorem and krulls principal. Finite commutative rings and their applications gilberto. The exercises at the ends of the chapters are mostly routine, with a few more.
Many areas of mathematics are used in coding theory, and we focus on the interplay between algebra and coding theory. Using this framework we construct an equivalence between the category of geometric linear codes over spec k and linear codes over k. Galois theory, commutative algebra, with applications to. Introduction in this paper we give a geometric version of the satake isomorphism sat. Finite frobenius rings as a setting for algebraic coding theory.
Schroeder, number theory in science and communication, springer, 1986, or indeed any book on. Idempotent is one of central notions in semigroup theory and algebra. Hence, in addition to being applicable to systems over the real or complex numbers, kalmans theory can be applied to systems over finite. A characterization of finite commutati ve rings roger a lperin and e loise h amann w e give a partial con verse to the w ellkno wn result. An important partial con verse of this which we use here is that. Algebraic complexities and algebraic curves over finite fields d. The main objects of study in this paper are the commutative separable algebras over a commutative ring.
Slides for my talk in the special session on algebraic coding theory at the joint mathematics meetings, san diego, california, january 7, 2002. Moreover, frobenius rings are the largest class of finite rings over which. Algebraic coding theory over finite commutative rings request. This book introduces algebraic coding theory over finite frobenius rings. In a finite commutative ring every nonzerodivisor is a unit.
For nonnoetherian rings, and also nonlocal rings, the dimension may be infinite, but noetherian local rings have finite dimension. On linear complimentary dual codes over finite commutative frobenius local rings ramakrishna bandi1. Matsumura covers the basic material, including dimension theory, depth, cohenmacaulay rings, gorenstein rings, krull rings and valuation rings. Some recent works on codes over such rings are 3, 7, 9, 22. It analyzes the following three properties of a code mainly. Does a finite commutative ring necessarily have a unity. Noncommutative separable algebras have been studied in 2. 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. Fulfillment by amazon fba is a service we offer sellers that lets them store their products in amazons fulfillment centers, and we directly pack, ship, and provide customer service for these products. Mceliece, finite fields for computer scientists and engineers, kluwer, 1987, m. Algebraic numbers, wileyinterscience, new york 1972. Geometric langlands duality and representations of algebraic groups over commutative rings by i. Quadratic gauss sums over finite commutative rings. Pdf commutative ring theory and applications download.
Is there a classification of finite commutative rings available. The notion of commutative algebraic theory was formulated in terms of monads by anders kock. Algebraic coding theory is an area of discrete applied mathematics that is concerned in part with developing errorcontrol codes and encodingdecoding procedures. Finite commutative rings and their applications request pdf. In the case r k is a field, the points correspond to the krational points. In addition to being an interesting and profound subject in its own right, commutative ring theory is important as a foundation for algebraic geometry and complex analytical geometry. Coding theory and applications, cim series in mathematical sciences, vol. Constructions of selfdual codes over finite commutative chain rings steven t. Commutative separable algebras have been studied in 1 and in 2, 6 where the main ideas are. Request pdf algebraic coding theory over finite commutative rings this book provides a selfcontained introduction to algebraic coding theory over finite frobenius rings. I ask because of the following theorem given in my lecture notes.
This book will be of interest to both professional and academic researchers in the fields of communication and coding theory. Algebraic coding theory over finite commutative rings steven t. Chudnovsky department of mathematics, columbia university, new york, new york 100. Finite commutative rings and their applications adds to the readers knowledge. Among others, necessary and sufficient conditions for the existence of selfdual, lcd, and hermitian dualcontaining. 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. Our definition of a code is motivated by the fact that every nonsingular algebraic variety x over the complex numbers c has a natural structure as a.
On the key equation over a commutative ring springerlink. I there were some results on codes over rings in the 1970s, but the real breakthrough came in 1994. Coding theory has its origins in the engineering problem of effective electronic communication where the alphabet is generally the binary field. Dougherty department of mathematics university of scranton scranton, pa 18510, usa. Finite frobenius rings as a setting for algebraic coding.
Hence we assume that the students have some basic background on ring theory and module theory. The object of this chapter is to combine these areas of coding theory by introducing and studying algebraic geometric codes over rings. Duality for modules over finite rings and applications to coding. Bif a finite commutative ring r satisfies that all of its hamming isometries.
Pdf finite commutative rings and their applications. Finite chain rings, besides their practical importance, are quite rich mathematical objects and so they have also theoretical attraction. Published for the conference board of the mathematical sciences by the american mathematical society, 2002. In this study, we investigate the algebraic structure of. If not, what are the best structure theorem that are known at present. Constacyclic codes over finite commutative semisimple rings. Computational methods in commutative algebra and algebraic geometry. The section on quasigalois rings presents new and unpublished results as well. Linear codes over finite chain ringsalgebraic theory. Thus the combinatorial topics may be enumerative in nature or involve matroids, polytopes, partially ordered sets, or finite geometries. Dedicated with respect and admiration to vera pless on the occasion of her retirement.
Hammons, kumar, calderbank, sloane, and sol e showed that important duality properties of certain. We provide a class of cyclic groups and a class of. Some standard terminology from algebraic coding theory is discussed in. The term algebraic coding theory denotes the subfield of coding theory where the properties of codes are expressed in algebraic terms and then further researched. A but we are particularly concerned with a when it has an algebraic structure. Finite commutative semisimple rings are direct sum of finite fields. This construction carries over to the wider context of monoidal monads, see tensor product of algebras over a commutative monad. They have connections in both geometry pappian hjelmslev planes and algebraic number theory quotient rings of algebraic integers.
The authors then introduce some applications of finite rings, in particular galois rings, to coding theory, using a solid algebraic and geometric theoretical background. On linear complimentary dual codes over finite commutative. Coding theory as pure mathematics page web dandre leroy. Constructions of selfdual codes over finite commutative chain rings. Homotopy invariance and quillens algebraic ktheory of rings 128 5. Commutative separable algebras have been studied in 1 and in 2, 6 where the main ideas are based on the classical galois theory of fields. It is the first to offer a comprehensive account on the subject. This book provides a selfcontained introduction to algebraic coding theory over finite frobenius rings. Request pdf algebraic coding theory over finite commutative rings this book provides a selfcontained introduction to algebraic coding theory over finite. We primarily investigate the the existence the lcd cyclic code of various lengths over.