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2018 | Book

b-Symbol Distance Distribution of Repeated-Root Cyclic Codes Read chapter Read first chapter

Homological and Combinatorial Methods in Algebra

SAA 4, Ardabil, Iran, August 2016

Editors: Prof. Ayman Badawi, Mohammad Reza Vedadi, Siamak Yassemi, Ahmad Yousefian Darani

Publisher: Springer International Publishing

Book Series : Springer Proceedings in Mathematics & Statistics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

Based on the 4th Seminar on Algebra and its Applications organized by the University of Mohaghegh Ardabili, this volume highlights recent developments and trends in algebra and its applications. Selected and peer reviewed, the contributions in this volume cover areas that have flourished in the last few decades, including homological algebra, combinatorial algebra, module theory and linear algebra over rings, multiplicative ideal theory, and integer-valued polynomials. Held biennially since 2010, SAA introduces Iranian faculty and graduate students to important ideas in the mainstream of algebra and opens channels of communication between Iranian mathematicians and algebraists from around the globe to facilitate collaborative research. Ideal for graduate students and researchers in the field, these proceedings present the best of the seminar’s research achievements and new contributions to the field.

Table of Contents

Frontmatter
b-Symbol Distance Distribution of Repeated-Root Cyclic Codes
Abstract
Symbol-pair codes, introduced by Cassuto and Blaum (Proc IEEE Int Symp Inf Theory, 988–992, 2010 [1]), have been raised for symbol-pair read channels. This new idea is motivated by the limitations of the reading process in high-density data storage technologies. Yaakobi et al. (IEEE Trans Inf Theory 62(4):1541–1551, 2016 [8]) introduced codes for b-symbol read channels, where the read operation is performed as a consecutive sequence of \(b>2\) symbols. In this paper, we come up with a method to compute the b-symbol-pair distance of two n-tuples, where n is a positive integer. Also, we deal with the b-symbol-pair distances of some kind of cyclic codes of length \(p^e\) over \({F}_{p^m}\).
Hojjat Mostafanasab, Esra Sengelen Sevim
Bhargava Rings Over Subsets
Abstract
Let D be an integral domain with quotient field K and let E be any nonempty subset of K. The Bhargava ring over E at \(x\in D\) is defined by \(\mathbb {B}_x(E,D):=\{f\in K[X]\mid f(xX+e)\in D[X], \ \forall e\in E\}\). This ring is a subring of the ring of integer-valued polynomials over E. This paper studies \(\mathbb {B}_x(E,D)\) for an arbitrary domain D. we provide information about its localizations and transfer properties, describe its prime ideal structure, and calculate its Krull and valuative dimensions.
I. Al-Rasasi, L. Izelgue
On Commutativity of Banach -Algebras with Derivation
Abstract
The aim of this paper is to apply purely ring theoretic results to discuss the commutativity of a Banach algebra and Banach \(^*\)-algebra via derivations. We prove that if \(\mathfrak {A}\) is a semiprime Banach algebra and \(\mathscr {G}\) a nonempty open subsets of \(\mathfrak {A}\) which admits a nonzero continuous linear derivation \(d:\mathfrak {A}\rightarrow \mathfrak {A}\) such that \(d([x^m-x,y])\in Z(\mathfrak {A})\) for each x in \(\mathscr {G}\) and an integer \(m=m(x)>1\), then \(\mathfrak {A}\) is commutative. Further, we discuss the commutativity of Banach \(^*\)-algebra. In particular, it is shown that either a semiprime Banach \(^*\)-algebra \(\mathfrak {A}\) with continuous involution and derivation is commutative or the set of \(x\in \mathfrak {A}\) for which \([d(x^k),d((x^k)^*)]\in Z(\mathfrak {A})\) for no positive integer \(k\ge 1\), is dense in \(\mathfrak {A}\). Finally, few more parallel results have been established about the commutativity of Banach and Banach \(^*\)-algebras.
Mohammad Ashraf, Bilal Ahmad Wani
An Application of Linear Algebra to Image Compression
Abstract
Nowadays the data are transmitted in the form of images, graphics, audio and video. These types of data require a lot of storage capacity and transmission bandwidth. Consequently, the theory of data compression becomes more significant for reducing the data redundancy in order to save more transfer and storage of data. In this context, this paper addresses the problem of the lossy compression of images. This proposed method is based on Block SVD Power Method that overcomes the disadvantages of Matlab’s SVD function. The quantitative and visual results are showing the superiority of the proposed compression method over those of Matlab’s SVD function and some different compression techniques in the state-of-the-art. In addition, the proposed approach is simple and can provide different degrees of error resilience, which gives, in a short execution time, a better image compression.
Khalid EL Asnaoui, Mohamed Ouhda, Brahim Aksasse, Mohammed Ouanan
Intuitionistic Fuzzy Group With Extended Operations
Abstract
In this paper we give an extension in the intuitionistic fuzzy frame of a low of a crisp group \((G,*)\) by means of the extended form of the Zadeh’s extension principle, And we build the conditions whose permit us to give an intuitionistic fuzzy group structure. Furthermore, we investigates some other properties for the intuitionistic fuzzy subgroups and homomorphisms for our set.
S. Melliani, I. Bakhadach, L. S. Chadli
Generalization of Quasi-modular Extensions
Abstract
Let K/k be a purely inseparable extension of characteristic \(p>0\). Let lm(K/k) and um(K/k) be the smallest extensions \(k \longrightarrow lm(K\)/\(k) \longrightarrow K\longrightarrow um(K\)/k) such that K/lm(K/k) and um(K/k)/k are modular. In this note, we continue to study the locus problem of lm(K/k) and um(K/k) relative to K/k. Thus improving ([3], Theorem 1.4), we show that lm(K/k) is nontrivial when K/k is of finite size, more precisely if K/k has a finite size and unbounded exponent, the same is true of K/lm(K/k). However, if K/k is of unbounded size, it may well be that we lose this property by obtaining lm(K/\(k)=K \). In the following, we will say that K/k is lq-modular (respectively, uq-modular) if lm(K/k)/k (respectively, um(K/k)/K) has an exponent. The first study of these two concepts devoted to the extensions of finite size is in [4, 6, 7]. However, the object of the present work consists to generalize the results of finite size to any extension. In particular, we treat the stability questions of the lq-modularity and the uq-modularity relative to inclusion, intersection, and product. Furthermore, we are interested by the questions about existence of the smallest extensions which preserve these concepts in the ascendant or descendant sense, and also to the questions of existence of the maximal subextensions (closures).
El Hassane Fliouet
A Class of Finite 2-groups G with Every Automorphism Fixing Elementwise
Abstract
The family \(G(m,n)=\langle x,y| x^2=(xy^2)^2=1,~y^{2^m}=(xy)^{2^{n}}\rangle \) of finite 2-groups will be introduced. The group G(m, n) has order \(2^{(m+n+1)}\), nilpotency class \(1+\max \{m,n\}\) and every automorphism of \(G=G(m,n)\) fixes \(G/\varPhi (G)\) elementwise and therefore Aut(G) is a 2-group. The parameterized presentation of \(G=G(m,n)\) is efficient as the Schur multiplicator of G is non-trivial.
Hossein Abdolzadeh, Reza Sabzchi
Fuzzy Rings and Fuzzy Polynomial Rings
Abstract
In this paper, we introduce the notion of a ring of fuzzy points, and study some basic properties and the relationship between this set and the classical ring R. We also define the fuzzy polynomial rings and fuzzy algebraic elements.
S. Melliani, I. Bakhadach, L. S. Chadli
On (Completely) Weak* Rad--Supplemented Modules
Abstract
In this paper, we establish various properties of weak* Rad-\(\oplus \)-supplemented modules and completely weak* Rad-\(\oplus \)-supplemented modules, which are the generalizations of \(\oplus \)–supplemented and Rad-\(\oplus \)-supplemented modules. Our main focus is to characterize the weak* Rad-\(\oplus \)-supplemented modules in terms of radical modules, modules having property \((p^*)\) and w–local modules.
Manoj Kumar Patel
When Is a Locally Free D–Module
Abstract
Let D be an integral domain with quotient field K, E a subset of K and X an indeterminate over K. The set of integer–valued polynomials on E is defined by \(\mathrm {Int}(E, D) = \{f \in K[X]:f (E) \subseteq D\}\). Clearly, \(\mathrm {Int}(E, D\)) is a subring of K[X] and \(\mathrm {Int}(D, D) = \mathrm {Int}(D)\), the ring of integer–valued polynomials over D. In this paper, we investigate some conditions under which \(\mathrm {Int}(E,D)\) is locally free, or at least flat, as a D–module. Particularly, we are interested in domains that are locally essential with subsets E residually cofinite.
Lahoucine Izelgue, Ali Tamoussit
Pairs of Rings Whose All Intermediate Rings Are G–Rings
Abstract
A G–ring is any commutative ring R with a nonzero identity such that the total quotient ring \(\mathbf {T}(R)\) is finitely generated as a ring over R. A G–ring pair is an extension of commutative rings \(A\hookrightarrow B\), such that any intermediate ring \(A\subseteq R\subseteq B\) is a G–ring. In this paper we investigate the transfer of the G–ring property among pairs of rings sharing an ideal. Our main result is a generalization of a theorem of David Dobbs about G–pairs to rings with zero divisors.
Lahoucine Izelgue, Omar Ouzzaouit
Weakly Finite Conductor Property in Amalgamated Algebra
Abstract
Let \(f: A{\longrightarrow } B\) be a ring homomorphism and J be an ideal of B. In this paper, we investigate the transfer of weakly finite conductor property in amalgamation of A with B along J with respect to f (denoted by \(A{\bowtie }^{f}J\)), introduced and studied by D’Anna, Finocchiaro and Fontana in 2009 (see D’Anna et al. (Commutative Algebra and Applications. Walter De Gruyter Publisher, Berlin, pp. 55–172, 2009), D’Anna et al. (J Pure Appl Algebra 214:1633–1641, 2010)). Our results generate original examples which enrich the current literature with new families of examples of nonfinite conductor weakly finite conductor rings.
Haitham El Alaoui
Coherence in Bi-amalgamated Algebras Along Ideals
Abstract
Let \(f: A\longrightarrow B\) and \(g: A\longrightarrow C\) be two ring homomorphisms and let J (resp., \(J'\)) be an ideal of B (resp., C) such that \(f^{-1}(J)=g^{-1}(J')\). In this paper, we investigate the transfer of the property of coherence in the bi-amalgamation of A with (B, C) along \((J,J')\) with respect to (f, g) (denoted by \(A\bowtie ^{f,g}(J,J'))\), introduced and studied by Kabbaj, Louartiti, and Tamekkante in 2013. We provide necessary and sufficient conditions for \(A\bowtie ^{f,g}(J,J')\) to be a coherent ring.
Mounir El Ouarrachi, Najib Mahdou
On the Set of Intermediate Artinian Subrings
Abstract
The paper contributes to the investigation of intermediate Artinian subrings between R and T, where \(R\hookrightarrow T\) is an extension of rings.
Driss Karim
Metadata
Title
Homological and Combinatorial Methods in Algebra
Editors
Prof. Ayman Badawi
Mohammad Reza Vedadi
Siamak Yassemi
Ahmad Yousefian Darani
Copyright Year
2018
Electronic ISBN
978-3-319-74195-6
Print ISBN
978-3-319-74194-9
DOI
https://doi.org/10.1007/978-3-319-74195-6

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