ON SOME TOTAL GRAPHS ON FINITE RINGS

Let $R$ be a commutative ring with non-zero identity, and $Z(R)$ be its set of all zero-divisors. The total graph of $R$, denoted by $T(\Gamma(R))$, is an undirected graph with all elements of $R$ as vertices, and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\in Z(R)$. In this article, we characterize, up to isomorphism, all of finite commutative rings whose total graphs have vertex-arboricity (arboricity) two or three. Also, we show that, for a positive integer $v$, the number of finite rings whose total graphs have vertex-arboricity (arboricity) $v$ is finite.

Linear dynamical systems over finite rings

On the ℓ-DLIPs of codes over finite commutative rings

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.

Let R be a commutative ring with Z(R), its set of zero-divisors and \mbox{Reg}(R), its set of regular elements. Total graph of R, denoted by T(\Gamma(R)), is the graph with all elements of R as vertices, and two distinct vertices x,y \in R, are adjacent in T(\Gamma(R)) if and only if x+y \in Z(R). In this paper, some properties of T(\Gamma(R)) have been investigated, where R is a finite commutative ring and a new upper bound for vertex-connectivity has been obtained in this case. Also, we have proved that the edge-connectivity of T(\Gamma(R)) coincides with the minimum degree if and only if R is a finite commutative ring such that Z(R) is not an ideal in R.

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.

Matrix product codes with rosenbloom-tsfasman metric

Given a finite (associative, unital) ring R, let K(R) denote the set of polynomials in R[x] that send each element of R to 0 under evaluation. We study K(R) and its elements. We conjecture that K(R) is a two-sided ideal of R[x] for any finite ring R, and prove the conjecture for several classes of finite rings (including commutative rings, semisimple rings, local rings, and all finite rings of odd order). We also examine a connection to sets of integer-valued polynomials.

Let [Formula: see text] be a subset of a commutative graded ring [Formula: see text]. The Cayley graph [Formula: see text] is a graph whose vertex set is [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. The Cayley sum graph [Formula: see text] is a graph whose vertex set is [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Let [Formula: see text] be the set of homogeneous elements and [Formula: see text] be the set of zero-divisors of [Formula: see text]. In this paper, we study [Formula: see text] (total graph) and [Formula: see text]. In particular, if [Formula: see text] is an Artinian graded ring, we show that [Formula: see text] is isomorphic to a Hamming graph and conversely any Hamming graph is isomorphic to a subgraph of [Formula: see text] for some finite graded ring [Formula: see text].

We study the compressed zero-divisor graph of a finite associative ring R. In particular, we describe commutative finite rings with compressed zero-divisor graphs of order 2. Moreover, we find all graphs of order 3 that are the compressed zero-divisor graphs of some finite rings.

Let F q \mathbb {F}_q denote the finite field of characteristic p p and order q q . Let Z q \mathbb {Z}_q denote the unramified extension of the p p -adic rational integers Z p \mathbb {Z}_p with residue field F q \mathbb {F}_q . Given two positive integers m , n m,n , define a box B m \mathcal B_m to be a subset of Z q n \mathbb {Z}_q^n with q n m q^{nm} elements such that B m \mathcal B_m modulo p m p^m is equal to ( Z q / p m Z q ) n (\mathbb {Z}_q/p^m \mathbb {Z}_q)^n . For a collection of nonconstant polynomials f 1 , … , f s ∈ Z q [ x 1 , … , x n ] f_1,\dots ,f_s\in \mathbb {Z}_q[x_1,\ldots ,x_n] and positive integers m 1 , … , m s m_1,\dots ,m_s , define the set of common zeros inside the box B m \mathcal B_m to be V = { X ∈ B m : f i ( X ) ≡ 0 mod p m i for all 1 ≤ i ≤ s } . \begin{equation*} V=\{X\in \mathcal B_m:\; f_i(X)\equiv 0\mod {p^{m_i}}\text { for all } 1\leq i\leq s\}. \end{equation*} It is an interesting problem to give the sharp estimates for the p p -divisibility of | V | |V| . This problem has been partially solved for the three cases: (i) m = m 1 = ⋯ = m s = 1 m=m_1=\cdots =m_s=1 , which is just the Ax-Katz theorem, (ii) m = m 1 = ⋯ = m s > 1 m=m_1=\cdots =m_s>1 , which was solved by Katz [Proc. Amer. Math. Soc. 137 (2009), pp. 4065–4075; Amer. J. Math. 93 (1971), pp. 485–499], Marshal and Ramage [Proc. Amer. Math. Soc. 49 (1975), pp. 35–38], and (iii) m = 1 m=1 , and m 1 , … , m s ≥ 1 m_1,\dots ,m_s\geq 1 , which was recently solved by Cao, Wan [Finite Fields Appl. 91 (2023), p. 25] and Grynkiewicz [ A generalization of the Chevalley-Warning and Ax-Katz theorems with a view towards combinatorial number theory , Preprint, arXiv: 2208.12895 , 2022]. Based on the multi-fold addition and multiplication of the finite Witt rings over F q \mathbb {F}_q , we investigate the remaining unconsidered case of m > 1 m>1 and m ≠ m j m\neq m_j for some 1 ≤ j ≤ s 1\leq j\leq s , and finally provide a complete answer to this problem.

Separations among the first order logic Ring(0,+, ∗) of finite residue class rings, its extensions with generalized quantifiers, and in the presence of a built-in order are shown, using algebraic methods from class field theory. These methods include classification of spectra of sentences over finite residue classes as systems of congruences, and the study of their h-densities over the set of all prime numbers, for various functions h on the natural numbers. Over ordered structures the logic of finite residue class rings and extensions are known to capture DLOGTIMEuniform circuit complexity classes ranging from AC to TC. Separating these circuit complexity classes is directly related to classifying the h-density of spectra of sentences in the corresponding logics of finite residue classes. We further give general conditions under which a logic over the finite residue class rings has a sentence whose spectrum has no h-density. One application of this result is that Supported by MICINN project TIN2011-27479-C04-03 (BASMATI), MINECO project TIN201457226-P (APCOM) and Gen. Cat. project SGR2014-890 (MACDA).

Let R be a commutative ring and Z(R) be its set of all zero-divisors. Anderson and Badawi [The total graph of a commutative ring, J. Algebra320 (2008) 2706–2719] introduced the total graph of R, denoted by TΓ(R), as the undirected graph with vertex set R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). Tamizh Chelvam and Asir [Domination in the total graph of a commutative ring, to appear in J. Combin. Math. Combin. Comput.] obtained the domination number of the total graph and studied certain other domination parameters of TΓ(R) where R is a commutative Artin ring. The intersection graph of gamma sets in TΓ(R) is denoted by ITΓ(R). Tamizh Chelvam and Asir [Intersection graph of gamma sets in the total graph, Discuss. Math. Graph Theory32 (2012) 339–354, doi:10.7151/dmgt.1611] initiated a study about the intersection graph ITΓ (ℤn) of gamma sets in TΓ(ℤn). In this paper, we study about ITΓ(R), where R is a commutative Artin ring. Actually we investigate the interplay between graph-theoretic properties of ITΓ(R) and ring-theoretic properties of R. At the first instance, we prove that diam (ITΓ(R)) ≤ 2 and gr (ITΓ(R)) ≤ 4. Also some characterization results regarding completeness, bipartite, cycle and chordal nature of ITΓ(R) are given. Further, we discuss about the vertex-transitive property of ITΓ(R). At last, we obtain all commutative Artin rings R for which ITΓ(R) is either planar or toroidal or genus two.

2 - Galois Fields

A cryptosystem for information exchange between subscribers based on surjective mappings of finite associative commutative rings with unity and systems of linear congruences over such rings is proposed. Algorithms for constructing surjective mappings of rings are presented, as well as a protocol for information exchange and the computational features of the means for implementing the protocol. The system does not require cumbersome calculations and construction of ring operation tables, and its stability is based on the combinatorial complexity of the set of surjective mappings and isomorphisms between finite rings of relatively small orders. The algorithms for solving systems of linear congruences that appear in the information exchange protocol are used to encrypt messages and have polynomial complexity. The operation of the cryptosystem is demonstrated with examples. Keywords: cryptography, symmetric cryptosystem, finite rings, isomorphism, system of linear congruences.