On Representability of a Finite Local Ring

Circulant matrices form an important class of matrices that have been continuously studied due to their nice algebraic structures and wide applications. In this study, we focus specifically on negacirculant matrices, which are known as extensions of the classical circulant matrices. The algebraic structures of the rings of n × n n\times n negacirculant matrices over finite fields and over finite commutative chain rings are presented. Subsequently, the algebraic structures and enumeration of the unit groups of such matrix rings are established. Additionally, the number of non-singular n × n n\times n negacirculant matrices with prescribed determinant is given in some cases. Conjectures and open problems are proposed as well as a brief discussion in the case where the underlying ring is a finite commutative ring with identity is also presented.

Symplectic graphs over finite local rings

In this paper, we study generalized Reed-Solomon codes (GRS codes) over commutative and noncommutative rings, we show that the classical Welch-Berlekamp and Guruswami-Sudan decoding algorithms still hold in this context, and we investigate their complexities. Under some hypothesis, the study of noncommutative GRS codes over finite rings leads to the fact that GRS codes over commutative rings have better parameters than their noncommutative counterparts. Also, GRS codes over finite fields have better parameters than their commutative rings counterparts. But we also show that given a unique decoding algorithm for a GRS code over a finite field, there exists a unique decoding algorithm for a GRS code over a truncated power series ring with a better asymptotic complexity. Moreover, we generalize a lifting decoding scheme to obtain new unique and list decoding algorithms designed to work when the base ring is, for example, a Galois ring or a truncated power series ring or the ring of square matrices over the latter ring.

It is very well known that algebraic structures have valuable applications in the theory of error-correcting codes. Blake [Codes over certain rings, Inform. and Control 20 (1972) 396–404] has constructed cyclic codes over ℤm and in [Codes over integer residue rings, Inform. and Control 29 (1975), 295–300] derived parity check-matrices for these codes. In [Linear codes over finite rings, Tend. Math. Appl. Comput. 6(2) (2005) 207–217]. Andrade and Palazzo present a construction technique of cyclic, BCH, alternant, Goppa and Srivastava codes over a local finite ring B. However, in [Encoding through generalized polynomial codes, Comput. Appl. Math. 30(2) (2011) 1–18] and [Constructions of codes through semigroup ring [Formula: see text] and encoding, Comput. Math. Appl. 62 (2011) 1645–1654], Shah et al. extend this technique of constructing linear codes over a finite local ring B via monoid rings [Formula: see text], where p = 2 and k = 1, 2, respectively, instead of the polynomial ring B[X]. In this paper, we construct these codes through the monoid ring [Formula: see text], where p = 2 and k = 1, 2, 3. Moreover, we also strengthen and generalize the results of [Encoding through generalized polynomial codes, Comput. Appl. Math.30(2) (2011) 1–18] and [Constructions of codes through semigroup ring [Formula: see text]] and [Encoding, Comput. Math. Appl.62 (2011) 1645–1654] to the case of k = 3.

Symplectic graphs over finite commutative rings

In 1963, Gilmer characterized all finite commutative rings with a cyclic group of units and, in 1967, Eldridge and Fischer generalized these results to rings with minimum condition. In the present paper these results are extended to semiperf ect rings and generalizations of the three theorems are obtained. It is shown that a semiperfect ring with cyclic group of units is finite and is either commutative or is the direct sum of a commutative ring and the 2 x 2 upper triangular matrix ring over the field of two elements. Let R be semiperfect with an abelian group of units. It is shown that R is finite if either the group of units is finite or the group of units is finitely generated and the Jacobson radical is nil. The proofs of all these results depend on our main theorem: The structure of a semiperfect ring R with an abelian group of units is described completely up to the structure of commutative local rings. (That is commutative rings with a unique maximal ideal.) The groups of units of these local rings are shown to be direct factors of the group of units of R.

For an element $x$ of a commutative ring $R$, let $\Gamma_x(R)$ be the graph whose vertices are the elements of $R$ that divide $x$ such that distinct vertices $r$ and $s$ are adjacent if and only if $rs=x$. If $x$ is a nonzero nonirreducible nonunit of an integral domain $R$, then $\Gamma_x^\mathcal{C}(R)$ is the graph whose vertices are the associate classes of divisors of $x$ that are neither units nor associates of $x$ such that distinct vertices $A$ and $B$ are adjacent if and only if $rs$ divides $x$ for some (and hence every) $r\in A$ and $s\in B$. The graphs $\Gamma_x(R)$ are considered when $R$ is a local Artinian principal ideal ring, and $\Gamma_x^\mathcal{C}(R)$ is examined when $R$ is a valuation domain. For example, it is shown that a finite local ring $R$ is a principal ideal ring if and only if there exists $x\in R$ such that every connected component of $\Gamma_x(R)$ is a star graph of a prescribed cardinality. Moreover, it is proved that an integral domain $R$ is a discrete valuation ring if and only if its collection of graphs $\Gamma_x^\mathcal{C}(R)$ consists precisely of a single-vertex graph, along with every (up to isomorphism) graph that is realizable as the compressed $0$-divisor graph of a local Artinian principal ideal ring. Certain graphs associated with partially ordered abelian groups have an essential role in the work.

No AccessTwisted Hessian curves over the ring 𝔽q[∈], ∈2 = 0Abdelâli Grini, Abdelhakim Chillali, Lhoussain El Fadil and Hakima MouanisAbdelâli GriniDepartment of Mathematics, Faculty of Science Dhar El Mahraz-Fez, S.M. Ben Abdellah University, Fez, Morocco, Abdelhakim ChillaliDepartment of Mathematics, Physical and Computer Sciences, FP, LSI, Taza, S.M. Ben Abdellah University, Fez, Morocco, Lhoussain El FadilDepartment of Mathematics, Faculty of Science Dhar El Mahraz-Fez, S.M. Ben Abdellah University, Fez, Morocco and Hakima MouanisDepartment of Mathematics, Faculty of Science Dhar El Mahraz-Fez, S.M. Ben Abdellah University, Fez, MoroccoPublished Online:November 28, 2022pp 181-189https://doi.org/10.1504/IJCAET.2023.127795PDF ToolsAdd to FavouritesDownload CitationsTrack Citations Share this article on social mediaShareShare onFacebookTwitterLinkedInReddit AboutAbstractThe goal of this work is to study some arithmetic proprieties of the twisted Hessian curves defined by a equation of type: aX3 + Y3 + Z3 = dXYZ on the local ring R2 = 𝔽q[X]/(X2), where p ≥ 5 is a prime number, q = pd and d ∈ ℕ*, such that -3 is not a square in 𝔽q. This paper consists of an introduction, section, and a conclusion. In the introduction, we review some fundamental arithmetic proprieties of finite local rings R2, which will be used in the remainder of this article. The section is devoted to the study the above mentioned twisted Hessian curves on these finite local rings for restriction to some specific characteristic p ≥ 5. Using these studies, we give essential properties and we define the group H2a,d, these properties, the classification of these elements and a bijection between the sets H2a,d and Ha0,d0 × 𝔽q, where Ha0,d0 is the twisted Hessian curve over the finite field 𝔽q.Keywordstwisted Hessian curves, finite over ring, cryptography, elliptic curves, local ring Previous article Next article FiguresReferencesRelatedDetails Volume 18Issue 1-32023 ISSN: 1757-2657eISSN: 1757-2665 HistoryPublished onlineNovember 28, 2022 Copyright © 2023 Inderscience Enterprises Ltd.Keywordstwisted Hessian curvesfinite over ringcryptographyelliptic curveslocal ringAuthors and AffiliationsAbdelâli Grini1 Abdelhakim Chillali2 Lhoussain El Fadil3 Hakima Mouanis4 1. Department of Mathematics, Faculty of Science Dhar El Mahraz-Fez, S.M. Ben Abdellah University, Fez, Morocco2. Department of Mathematics, Physical and Computer Sciences, FP, LSI, Taza, S.M. Ben Abdellah University, Fez, Morocco3. Department of Mathematics, Faculty of Science Dhar El Mahraz-Fez, S.M. Ben Abdellah University, Fez, Morocco4. Department of Mathematics, Faculty of Science Dhar El Mahraz-Fez, S.M. Ben Abdellah University, Fez, MoroccoPDF download

Let $R$ be a finite commutative ring of odd characteristic and let $V$ be a free $R$-module of finite rank. We classify symmetric inner products defined on $V$ up to congruence and find the number of such symmetric inner products. Additionally, if $R$ is a finite local ring, the number of congruent symmetric inner products defined on $V$ in each congruence class is determined.

It was shown by C. Wickham [12] that “for a fixed positive integer g, there are finitely many isomorphism classes of finite commutative rings whose zero-divisor graph has genus g.” In this note, we give a short direct proof for this result. Moreover, we show that, if the zero-divisor graph of a commutative ring R has finite genus g, then either g = 0 or R is a finite ring. This immediately generalizes Wickham's theorem to arbitrary (not necessary finite) commutative rings.

On the ℓ-DLIPs of codes over finite commutative rings

Several different generalizations of finite factorization domains (i.e., integral domains where every nonzero nonunit has only finitely many divisors up to associates) have been defined for commutative rings with zero divisors. We study these notions in the context of commutative monoid rings with zero divisors, utilizing semigroup theory to simultaneously generalize and extend many past results about “finite factorization” properties in commutative polynomial rings. Along the way, we expand upon the general theory of factorization in commutative rings with zero divisors, providing new characterizations and results about several kinds of “finite factorization rings.”

The Harary Index is an important topological parameter for examining the structure of a graph. This work presents a quantitative analysis of the structural features of zero-divisor graph using the Harary Index. In this research article, we compute the Harary Index for the zero-divisor graph of the ring of upper triangular matrices over a finite commutative ring and non commutative ring. In this study, we have computed the Harary index for directed and undirected zero-divisor graph obtained from upper triangular matrix [Formula: see text].

Linear dynamical systems over finite rings

Matrix product codes with rosenbloom-tsfasman metric