Finite Rings Research Articles

Bijective soft rings with applications

In recent years, soft sets have been widely used in many important decision-making real-life problems. In this paper, observing the usage of soft sets in such kind of vital problems, we have introduced the bijective-unitary bijective soft rings. Firstly, we have defined and exemplified a bijective soft ring and a unitary bijective soft ring. Moreover, we have presented some applications of bijective soft rings. We have shown the usage of bijective soft rings in coding theory. In this context, we have observed that by obtaining a bijective soft ring over a finite ring, we have a coding matrix to encode a given set of messages. Besides these applicable results, we have also obtained some relations between bijective soft and classical rings.

On circulant involutory and orthogonal MDS matrices over finite commutative rings

On circulant involutory and orthogonal MDS matrices over finite commutative rings

Full Classification of Finite Singleton Local Rings

The main objective of this article is to classify all finite singleton local rings, which are associative rings characterized by a unique maximal ideal and a distinguished basis consisting of a single element. These rings are associated with four positive integer invariants p,n,s, and t, where p is a prime number. In particular, we aim to classify these rings and count them up to isomorphism while maintaining the same set of invariants. We have found interesting cases of finite singleton local rings with orders of p6 and p7 that hold substantial importance in the field of coding theory.

Solving systems of algebraic equations over finite commutative rings and applications

AbstractSeveral problems in algebraic geometry and coding theory over finite rings are modeled by systems of algebraic equations. Among these problems, we have the rank decoding problem, which is used in the construction of public-key cryptosystems. A finite chain ring is a finite ring admitting exactly one maximal ideal and every ideal being generated by one element. In 2004, Nechaev and Mikhailov proposed two methods for solving systems of polynomial equations over finite chain rings. These methods used solutions over the residue field to construct all solutions step by step. However, for some types of algebraic equations, one simply needs partial solutions. In this paper, we combine two existing approaches to show how Gröbner bases over finite chain rings can be used to solve systems of algebraic equations over finite commutative rings. Then, we use skew polynomials and Plücker coordinates to show that some algebraic approaches used to solve the rank decoding problem and the MinRank problem over finite fields can be extended to finite principal ideal rings.

On vertex connectivity of zero-divisor graphs of finite commutative rings

On vertex connectivity of zero-divisor graphs of finite commutative rings

Solving modular cubic equations with Coppersmith's method

Several cryptosystems based on Elliptic Curve Cryptography such as KMOV and Demytko process the message as a point M=(x0,y0) of an elliptic curve with an equation of the form y2≡x3+ax+b(modn) over a finite field when n is a prime number, or over a finite ring when n=pq is an RSA modulus. Other systems use singular cubic curves such as y2≡x3+ax2(modn) and y2+axy≡x3(modn). In this paper, we present a method to find the small solutions of the former modular cubic equations. Our method is based on Coppersmith's technique and enables one to find the solutions (x0,y0) when |x0|3|y0|2 is smaller than the modulus.

General fractal dimensions of graphs of products and sums of continuous functions and their decompositions

This study takes a broad approach to the fractal geometry problem and proposes an intrinsic definition of the general box dimensions and the general Hausdorff and packing dimensions by taking into account sums of the type ∑ih−1(sg(ri)) for some prescribed functions h, g and for all positive real s. Our primary aim is to conduct a more comprehensive exploration of the fractal dimensions exhibited by graphs resulting from the products and sums of two continuous functions. This article is entirely devoted to the investigation of this specific problem domain. In the course of our research, we furnish explicit formulas for computing both the general upper box dimension and the general lower box dimension associated with the graphs formed by the products and sums of two continuous functions. Additionally, we shed light on the decomposition of continuous functions into the sum of two continuous functions, employing the framework of the general lower and upper box dimensions. Furthermore, we establish an upper bound for the general upper box dimension applicable to each element within a finite ring of polynomials derived from continuous functions defined over the real number field R. Our inquiry also extends to the examination of general Hausdorff and packing dimensions for the graphs generated through both the summation and multiplication of two continuous functions.

On L(2, 1)-labeling of zero-divisor graphs of finite commutative rings

On L(2, 1)-labeling of zero-divisor graphs of finite commutative rings

Abelian and consta-Abelian polyadic codes over affine algebras with a finite commutative chain coefficient ring

This paper studies Abelian and consta-Abelian polyadic codes over rings defined as affine algebras over chain rings. For this purpose, we use the classical construction via splittings and multipliers of the underlying Abelian group. We also derive some results on the structure of the associated polyadic codes and the number of codes under these conditions.

Idempotent and nilpotent elements in octonion rings over Z

In this paper, we show that the set O/Zp, where p is a prime number, does not form a skew field and discuss idempotent and nilpotent elements in the (finite) ring O/Zp. We provide examples and establish conditions for idempotency and nilpotency. Mathematics Subject Classification (2010): 15A33, 15A30, 20H25, 15A03. Received 27 July 2021; Accepted 14 December 2021

REPRODUCED PRINCIPAL IDEAL DOMAIN ON GENERAL HYPERRING Zp^nq^m

Every classical algebra is a set equipped with binary operations that operate under certain axiom principles. The generalization of classical algebras to hyperalgebras has been created with the aim of generalizing operations to hyperoperations that apply to specific subject principles. This paper introduces the concept of reproduced general hyperrings as a generalization of rings and investigates and analyzes some of their essential properties. This study defines the notation of reproduced hyperideals in reproduced general hyperrings, consider the ideals of finite rings and obtain the finite and cyclic hyperideals. In the endl, we introduce and show that a principal Ideal domain finite reproduced general hyperring is Ideal-absorbing.

Solving polynomial systems over non-fields and applications to modular polynomial factoring

We study the problem of solving a system of m polynomials in n variables over the ring of integers modulo a prime-power pk. The problem over finite fields is well studied in varied parameter settings. For small characteristic p=2, Lokshtanov et al. (SODA'17) initiated the study, for degree d=2 systems, to improve the exhaustive search complexity of O(2n)⋅poly(m,n) to O(20.8765n)⋅poly(m,n); which currently is improved to O(20.6943n)⋅poly(m,n) in Dinur (SODA'21). For large p but constant n, Huang and Wong (FOCS'96) gave a randomized poly(d,m,log⁡p) time algorithm. Note that for growing n, system-solving is known to be intractable even with p=2 and degree d=2.We devise a randomized poly(d,m,log⁡p)-time algorithm to find a root of a given system of m integral polynomials of degrees bounded by d, in n variables, modulo a prime power pk; when n+k is constant. In a way, we extend the efficient algorithm of Huang and Wong (FOCS'96) for system-solving over Galois fields (i.e., characteristic p) to system-solving over Galois rings (i.e., characteristic pk); when k>1 is constant. The challenge here is to find a lift of singularFp-roots (exponentially many); as there is no efficient general way known in algebraic-geometry for resolving singularities.Our algorithm has applications to factoring univariate polynomials over Galois rings. Given f∈Z[x] and a prime-power pk (k≥2), finding factors of fmodpk has a curious state-of-the-art. It is solved for large k by p-adic factoring algorithms (von zur Gathen, Hartlieb, ISSAC'96); but unsolved for small k. In particular, no nontrivial factoring method is known for k≥5 (Dwivedi, Mittal, Saxena, ISSAC'19). One issue is that degree-δ factors of f(x)modpk could be exponentially many, as soon as k≥2. We give the first randomized poly(deg⁡(f),log⁡p)-time algorithm to find a degree-δ factor of f(x)modpk, when k+δ is constant. Our method has potential application in algebraic coding theory. In particular, extending algebraic geometric and Reed-Solomon codes to Galois rings could enable new and improved bounds on their underlying efficiency parameters.

Determinants of tridiagonal matrices over some commutative finite chain rings

Abstract Diagonal matrices and their generalization in terms of tridiagonal matrices have been of interest due to their nice algebraic properties and wide applications. In this article, the determinants of tridiagonal matrices over a finite field F q {{\mathbb{F}}}_{q} and a commutative finite chain ring R R are studied. The main focus is the enumeration of tridiagonal matrices with prescribed determinant. The number of tridiagonal matrices with prescribed determinant over F q {{\mathbb{F}}}_{q} and the number of non-singular tridiagonal matrices with prescribed determinant over R R are completely determined. For singular tridiagonal matrices with prescribed determinant over R R , bounds on the number of such matrices with prescribed determinant are given. Subsequently, the number of some special tridiagonal matrices with prescribed determinant over F q {{\mathbb{F}}}_{q} and R R is presented.

Matrices having nonzero outer inverses

It is well known that every nonzero von Neumann regular $m\times n$-matrix $A$ over an arbitrary ring $R$ has a nonzero outer inverse $n\times m$-matrix $B$ in the sense that $B=BAB$. Generalizing previous work on von Neumann regular matrices, the matrices having nonzero outer inverses over semiperfect rings are characterized as the matrices having some entry outside the Jacobson radical of $R$. Such matrices over finite semiperfect rings and finite commutative rings are counted, and several applications are given.

Commuting and product-zero probability in finite rings

Let [Formula: see text] be the probability that two random elements of a finite ring R commute and [Formula: see text] the probability that the product of two random elements in R is zero. We show that if [Formula: see text], then there exists a Lie-ideal D in the Lie-ring [Formula: see text] with [Formula: see text]-bounded index and with [Formula: see text] of [Formula: see text]-bounded order. If [Formula: see text], then there exists an ideal D in R with [Formula: see text]-bounded index and [Formula: see text] of [Formula: see text]-bounded order. These results are analogous to the well-known theorem of Neumann on the commuting probability in finite groups.

Group structure of elliptic curves over ℤ/Nℤ

Abstract We characterize the possible groups E ( Z ∕ N Z ) E\left({\mathbb{Z}}/N{\mathbb{Z}}) arising from elliptic curves over Z ∕ N Z {\mathbb{Z}}/N{\mathbb{Z}} in terms of the groups E ( F p ) E\left({{\mathbb{F}}}_{p}) , with p p varying among the prime divisors of N N . This classification is achieved by showing that the infinity part of any elliptic curves over Z ∕ p e Z {\mathbb{Z}}/{p}^{e}{\mathbb{Z}} is a Z ∕ p e Z {\mathbb{Z}}/{p}^{e}{\mathbb{Z}} -torsor, of which a generator is exhibited. As a first consequence, when E ( Z ∕ N Z ) E\left({\mathbb{Z}}/N{\mathbb{Z}}) is a p p -group, we provide an explicit and sharp bound on its rank. As a second consequence, when N = p e N={p}^{e} is a prime power and the projected curve E ( F p ) E\left({{\mathbb{F}}}_{p}) has trace one, we provide an isomorphism attack to the elliptic curve discrete logarithm problem, which works only by means of finite ring arithmetic.

Fuchs’ problem for linear groups

Which groups can occur as the group of units in a ring? Such groups are called realizable. Though the realizable members of several classes of groups have been determined (e.g., cyclic, odd order, alternating, symmetric, finite simple, indecomposable abelian, and dihedral), the question remains open. The general linear groups are realizable by definition: they are the units in the corresponding matrix rings. In this paper, we study the realizability of two closely related linear groups, the special linear groups and the affine general linear groups. We determine which special linear groups of degree 2 over a finite field are realizable by a finite ring, and we determine which affine general linear groups of degree 1 over a cyclic group are realizable by a finite ring. We also give partial results for certain linear groups of other degrees and for rings of characteristic zero.

On the unitary one matching Bi-Cayley graph over finite rings

Let [Formula: see text] be a finite ring (with nonzero identity) and let [Formula: see text] denote the unitary one-matching bi-Cayley graph over [Formula: see text] In this paper, we calculate the chromatic, edge chromatic, clique and independent numbers of [Formula: see text] and we show that the graph [Formula: see text] is a strongly regular graph if and only if [Formula: see text]. Also, we study the perfectness of [Formula: see text] Moreover, we prove that if [Formula: see text] and [Formula: see text] then [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are Jacobson radicals of [Formula: see text] and [Formula: see text], respectively. Furthermore, for a finite field [Formula: see text] with [Formula: see text] and a ring [Formula: see text], we prove that if [Formula: see text] where [Formula: see text] is an integer, then [Formula: see text] and so [Formula: see text] is a semisimple ring.

Determinants of Arrowhead Matrices over Finite Commutative Chain Rings

Arrowhead matrices have attracted attention due to their rich algebraic structures and numerous applications. In this paper, we focus on the enumeration of n × n arrowhead matrices with prescribed determinant over a finite field Fq and over a finite commutative chain ring R. The number of n × n arrowhead matrices over Fq of a fixed determinant a is determined for all positiveintegers n and for all elements a ∈ Fq. As applications, this result is used in the enumeration of n × n non-singular arrowhead matrices with prescribed determinant over R. Subsequently, some bounds on the number of n × n singular arrowhead matrices over R of a fixed determinant are given. Finally, some open problems are presented.

Factorization properties of quotients of polynomial and power series rings by monomial ideals

Suppose that K is a field, S = K [ x 1 , … , x n ] or S = K [ [ x 1 , … , x n ] ] and I is a monomial ideal of S. We study factorization properties of the ring R = S / I . In particular, we present conditions equivalent to R being présimplifiable, a bounded factorization ring, a finite factorization ring or a unique factorization ring.