Finite Rings Research Articles

Extended Total Graph Associated with Finite Commutative Rings

Extended Total Graph Associated with Finite Commutative Rings

The implementation of algebraic complex lookup tables over non-chain Galois ring extensions and Henon map in multimedia security

Abstract Image encryption is crucial for web-based data storage and transmission. Complex algebraic structures play a vital role in providing unique features and binary operations. However, current algebraic-based techniques face challenges due to limited key space. To tackle this issue, our study uniquely connects the algebraic structures with a chaotic map. The study introduces a complex non-chain Galois ring structure and a 12-bit substitution box for image substitution. An affine map is utilized to permute image pixels, and the 12-bit substitution box is uniquely mapped to a Galois field for encryption. A two-dimensional Henon map is employed to generate different keys for the XOR operation, resulting in an encrypted image. The resilience of the scheme against various attacks is evaluated using statistical, differential, and quality measures, showcasing its effectiveness against well-known attacks.

Frobenius Local Rings of Order p4m

Suppose R is a finite commutative local ring, then it is known that R has four positive integers p,n,m,k called the invariants of R, where p is a prime number. This paper investigates the structure and classification up to isomorphism of local rings with residue field Fpm and of length 4. Specifically, it gives a comprehensive characterization of Frobenius local rings of order p4m. Furthermore, we provide a detailed enumeration of the classes of all such rings with respect to their invariants p,n,m,k. Finite Frobenius rings are particularly advantageous for coding theory. This suitability arises from the fact that two classical theorems by MacWilliams, the Extension Theorem and the MacWilliams relations for symmetrized weight enumerators, can be generalized from finite fields to finite Frobenius rings.

Unit Groups of a Finite Extension of Galois Rings

Let $R_{o}$ be a Galois ring. It is well known that every element of $R_{o}$ is either a zero divisor or a unit. Galois rings are building blocks of completely primary finite rings which have yielded interesting results towards classification of finite rings. Recent studies have revealed that every finite commutative ring is a direct sum of completely primary finite rings. In fact, extensive account of finite rings have been given in the recent past. However, the classification of finite rings into well known structures still remains an open problem. For instance, the structure of the group of units of $R_{o}$ is known and some results have been obtained on the structure of its zero divisor graphs. In this paper, we construct a finite extension of $R_{o}$ (a special class of completely primary finite rings) and classify its group of units for all the characteristics.

Unit Groups of a Finite Extension of Galois Rings

Let $R_{o}$ be a Galois ring. It is well known that every element of $R_{o}$ is either a zero divisor or a unit. Galois rings are building blocks of completely primary finite rings which have yielded interesting results towards classification of finite rings. Recent studies have revealed that every finite commutative ring is a direct sum of completely primary finite rings. In fact, extensive account of finite rings have been given in the recent past. However, the classification of finite rings into well known structures still remains an open problem. For instance, the structure of the group of units of $R_{o}$ is known and some results have been obtained on the structure of its zero divisor graphs. In this paper, we construct a finite extension of $R_{o}$ (a special class of completely primary finite rings) and classify its group of units for all the characteristics.

Enumeration of matrices with single unit-entry rows over finite commutative rings

Let [Formula: see text] be a finite commutative ring with identity. In this paper, formal expressions of the number of [Formula: see text] matrices over [Formula: see text] of rank [Formula: see text] and the number of invertible matrices over [Formula: see text] are presented. The number of matrices over [Formula: see text] with a given rank and a given number of single unit-entry rows, rows in which a single entry is a unit and all other entries are zero, is finally determined.

Degree-based topological indices of the idempotent graph of the ring [formula omitted]

Let R be a finite commutative ring with a non-zero identity, and Id(R) be the set of idempotent elements of R. The idempotent graph of R, denoted by GId(R), is a simple undirected graph with all elements of R as vertices, and two distinct vertices u, v are adjacent if and only if u+v∈Id(R). In this paper, we consider the idempotent graph of the ring Zn and investigate some degree-based topological indices, such as the general sum-connectivity index, the general Randić index, the general Zagreb index, and the Sombor index of that graph by considering the M-polynomial of the graph.

Relative injective modules, superstability and noetherian categories

We study classes of modules closed under direct sums, [Formula: see text]-submodules and [Formula: see text]-epimorphic images where [Formula: see text] is either the class of embeddings, RD-embeddings or pure embeddings. We show that the [Formula: see text]-injective modules of theses classes satisfy a Baer-like criterion. In particular, injective modules, RD-injective modules, pure injective modules, flat cotorsion modules and [Formula: see text]-torsion pure injective modules satisfy this criterion. The argument presented is a model theoretic one. We use in an essential way stable independence relations which generalize Shelah’s non-forking to abstract elementary classes. We show that the classical model theoretic notion of superstability is equivalent to the algebraic notion of a noetherian category for these classes. We use this equivalence to characterize noetherian rings, pure semisimple rings, perfect rings and finite products of finite rings and artinian valuation rings via superstability.

Structures of finite fields of primes and the Euclidean square roots of unity in the finite rings

Structures of finite fields of primes and the Euclidean square roots of unity in the finite rings

Some Results for Double Cyclic Codes over Fq+vFq+v2Fq.

Let Fq be a finite field with an odd characteristic. In this paper, we present a new result about double cyclic codes over a finite non-chain ring. Specifically, we study the double cyclic code over Fq+vFq+v2Fq with v3=v, which is isomorphic to Fq×Fq×Fq. This study mainly involves generator polynomials and generator matrices. The generating polynomial of the dual code is also obtained. We show the relationship between the generating polynomials of the double cyclic codes and those of their dual codes. Finally, as an application of these results, we construct some optimal codes over F3.

Linear Codes over $\mathbb{Z}_{p}\mathcal{R}_{1} \mathcal{R}_{2}$ and their Applications

In the paper, we explore the simplex and MacDonald codes over the finite ring $\mathbb{Z}_{p}\mathcal{R}_{1} \mathcal{R}_{2}$. Our investigation focuses on the unique properties of these codes, with the particular attention to their weight distributions and Gray images. The weight distribution is a crucial aspect as it provides insights into the error-detection and error-correction capabilities of the codes. Gray images play a significant role in understanding the structure and behavior of these codes. By examining the dual Gray images of simplex and MacDonald codes over $\mathbb{Z}_{p}\mathcal{R}_{1} \mathcal{R}_{2}$, we aim to develop efficient secret sharing schemes. These schemes benefit from the inherent properties of the codes, such as minimal weight and redundancy, which are essential for secure and reliable information sharing. Understanding the access structure of these schemes is vital, as it determines which subsets of participants can reconstruct the secret. Our study draws on various properties to elucidate this access structure, ensuring that the schemes are secure and efficient. Through this comprehensive analysis, we contribute to the field of coding theory by demonstrating how simplex and MacDonald codes over $\mathbb{Z}_{p}\mathcal{R}_{1} \mathcal{R}_{2}$ can be effectively utilized in cryptographic applications, particularly in designing robust and reliable secret sharing mechanisms.

Improvement of Blowfish encryption algorithm based on Galois ring for electro-optical satellite images

Improvement of Blowfish encryption algorithm based on Galois ring for electro-optical satellite images

A New Method for Constructing Self-Dual Codes over Finite Commutative Rings with Characteristic 2

In this work, we present a new method for constructing self-dual codes over finite commutative rings R with characteristic 2. Our method involves searching for k×2k matrices M over R satisfying the conditions that its rows are linearly independent over R and MM⊤=α⊤α for an R-linearly independent vector α∈Rk. Let C be a linear code generated by such a matrix M. We prove that the dual code C⊥ of C is also a free linear code with dimension k, as well as C/Hull(C) and C⊥/Hull(C) are one-dimensional free R-modules, where Hull(C) represents the hull of C. Based on these facts, an isometry from Rx+Ry onto R2 is established, assuming that x+Hull(C) and y+Hull(C) are bases for C/Hull(C) and C⊥/Hull(C) over R, respectively. By utilizing this isometry, we introduce a new method for constructing self-dual codes from self-dual codes of length 2 over finite commutative rings with characteristic 2. To determine whether the matrix MM⊤ takes the form of α⊤α with α being a linearly independent vector in Rk, a necessary and sufficient condition is provided. Our method differs from the conventional approach, which requires the matrix M to satisfy MM⊤=0. The main advantage of our method is the ability to construct nonfree self-dual codes over finite commutative rings, a task that is typically unachievable using the conventional approach. Therefore, by combining our method with the conventional approach and selecting an appropriate matrix construction, it is possible to produce more self-dual codes, in contrast to using solely the conventional approach.

Finite commutative rings whose line graphs of comaximal graphs have genus at most two

Let $R$ be a ring with identity. The comaximal graph of $R$, denoted by $\Gamma(R)$, is a simple graph with vertex set $R$ and two different vertices $a$ and $b$ are adjacent if and only if $aR+bR=R$. Let $\Gamma_{2}(R)$ be a subgraph of $\Gamma(R)$ induced by $R\backslash\{U(R)\cup J(R)\}$. In this paper, we investigate the genus of the line graph $L(\Gamma(R))$ of $\Gamma(R)$ and the line graph $L(\Gamma_{2}(R))$ of $\Gamma_2(R)$. All finite commutative rings whose genus of $L(\Gamma(R))$ and $L(\Gamma_{2}(R))$ are 0, 1, 2 are completely characterized, respectively.

Projections of Finite Rings

Projections of Finite Rings

Some New Results on S-prime Ideals of a Finite Commutative Ring as S-meet Semilattice

Let ℜ be the finite commutative ring with unity and <i>I<sub>s</sub></i> be the S-prime ideal of a ring ℜ. The set <i>L<sub>s</sub></i> forms a partially ordered set (poset) by the subset relation. Initially, the interplay of the semilattice theoretic properties of a poset with the ring theoretic properties are studied with suitable examples. The number of maximal chain of a poset is compared with the number of prime ideals of a ring. It is proved that every maximal element of a poset is the prime ideal of a ring. A prime order ring is shown as a lattice. If the order of the ring is the product of two primes, then the trivial ideal is expressed as the meet of every pair of a poset. Further, the cardinality of the poset is determined in terms of the divisors of the order of the ring . A new meet-semilattice called the S-meet semilattice (<i>L<sub>s</sub></i>, ⋀, ⊆) is defined and the generalized Hasse diagrams of the S-meet semilattice of a ring of prime powers, product of prime powers are drawn in this paper in order to find the properties of S-meet semilattice. Finally, the ideals, the prime ideals and the maximal ideals of the S-meet semilattice are described in terms of the down-sets of S-meet semilattice where the results are listed with an example at the end.

Binding Number Bounds of Zero Divisor Graphs of Classes of Completely Primary Finite Rings

The binding number of a graph is an important graph parameter which measures the distribution of the size of the graph and its related properties including toughness, rapture degree, scattering number and its integrity. For complete graphs G ≃ Kn obtained from commutative finite ring, some results exist on the bounds of binding numbers. In this paper, we consider an incomplete but connected zero divisor graph Γ(R) associated with a class of completely primary finite ring R and use standard procedures to compute the binding number bounds, the average binding number and related graph parameters.

Analysis of Adjacency, Laplacian and Distance Matrices of Zero Divisor Graphs of 4-Radical Zero Completely Primary Finite Rings

Analysis of Adjacency, Laplacian and Distance Matrices of Zero Divisor Graphs of 4-Radical Zero Completely Primary Finite Rings

Self dual and MHDR dual cyclic codes over finite chain rings

Self dual and MHDR dual cyclic codes over finite chain rings

New Constructions of Approximately Mutually Unbiased Bases by Character Sums over Galois Rings

New Constructions of Approximately Mutually Unbiased Bases by Character Sums over Galois Rings