Comaximal Factorization Graphs in Integral Domains

Communications in Algebra, 2011

This article examines the connections between the factorization properties of a domain, eg. UFD, FFD, and the domain's irreducible divisor graphs. In particular, we show that although there are some nice correlations between the properties of the domain D and the set of irreducible divisor graphs {G(x) : x ∈ D * \U (D)} when D is a Finite Factorization Domain, it is very unlikely that any information about the domain D can be gleaned from the collection {G(x) : x ∈ D * \U (D)} when D is not a Finite Factorization Domain.

Canadian Mathematical Bulletin, 2014

It is well known that the factorization properties of a domain are reflected in the structure of its group of divisibility. The main theme of this paper is to introduce a topological/graph-theoretic point of view to the current understanding of factorization in integral domains. We also show that connectedness properties in the graph and topological space give rise to a generalization of atomicity.

arXiv: Commutative Algebra, 2015

The compressed zero-divisor graph $\Gamma_C(R)$ associated with a commutative ring $R$ has vertex set equal to the set of equivalence classes $\{ [r] \mid r \in Z(R), r \neq 0 \}$ where $r \sim s$ whenever $ann(r) = ann(s)$. Distinct classes $[r],[s]$ are adjacent in $\Gamma_C(R)$ if and only if $xy = 0$ for all $x \in [r], y \in [s]$. In this paper, we explore the compressed zero-divisor graph associated with quotient rings of unique factorization domains. Specifically, we prove several theorems which exhibit a method of constructing $\Gamma(R)$ for when one quotients out by a principal ideal, and prove sufficient conditions for when two such compressed graphs are graph-isomorphic. We show these conditions are not necessary unless one alters the definition of the compressed graph to admit looped vertices, and conjecture necessary and sufficient conditions for two compressed graphs with loops to be isomorphic when considering any quotient ring of a unique factorization domain.

Malaya Journal of Matematik

We introduce the prime graph of the product ring R 1 ×R 2 where R 1 , R 2 are integral domains, which is an extension of study on prime graph of an integral domain. We prove that, if R 1 , R 2 are two integral domains, the graph obtained by removing the isolated vertices from PG(R 1 ×R 2) is a bipartite graph. We obtain some consequences.

Advances in Mathematics: Scientific Journal, 2020

Let R be a commutative ring and let Γ(Z n) be the zero divisor graph of a commutative ring R, whose vertices are non-zero zero divisors of Z n , and such that the two vertices u, v are adjacent if n divides uv. In this paper, we introduce the concept of Decomposition of Zero Divisor Graph in a commutative ring and also discuss some special cases of Γ(Z 2 2 p), Γ(Z 3 2 p), Γ(Z 5 2 p), Γ(Z 7 2 p) and Γ(Z p 2 q).

Malaysian Journal of Mathematical Sciences, 2023

The study of rings and graphs has been explored extensively by researchers. To gain a more effective understanding on the concepts of the rings and graphs, more researches on graphs of different types of rings are required. This manuscript provides a different study on the concepts of commutative rings and undirected graphs. The non-zero divisor graph, Γ(R) of a ring R is a simple undirected graph in which its set of vertices consists of all non-zero elements of R and two different vertices are joint by an edge if their product is not equal to zero. In this paper, the commutative rings are the ring of integers modulo n where n = 8k and k ≤ 3. The zero divisors are found first using the definition and then the non-zero divisor graphs are constructed. The manuscript explores some properties of non-zero divisor graph such as the chromatic number and the clique number. The result has shown that Γ(Z 8k) is perfect.

Applied Mathematics, 2013

      such as connectivity, diameter, girth, clique numbers and planarity. We also study the cozero-divisor graphs of the direct products of two arbitrary commutative rings.

2020

Let R be a ring, we associate a simple graph Φ(R) to R, with vertices V (R) = R\{0, 1,−1}, where distinct vertices x, y ∈ V (R) are adjacent if and only if either xy ̸= 0 or yx ̸= 0. In this paper, we prove that if Φ(R) is connected such that R Z2×Z4 then the diameter of Φ(R) is almost 2. Also, we will pay specific attention to investigate the connectivity of certain rings such that, the ring of integers modulo n,Zn is connected, reduced ring and matrix ring.

Proceedings - Mathematical Sciences, 2018

Let R be a commutative ring with a nonzero identity element. For a natural number n, we associate a simple graph, denoted by n R , with R n \{0} as the vertex set and two distinct vertices X and Y in R n being adjacent if and only if there exists an n × n lower triangular matrix A over R whose entries on the main diagonal are nonzero and one of the entries on the main diagonal is regular such that X T AY = 0 or Y T AX = 0, where, for a matrix B, B T is the matrix transpose of B. If n = 1, then n R is isomorphic to the zero divisor graph (R), and so n R is a generalization of (R) which is called a generalized zero divisor graph of R. In this paper, we study some basic properties of n R. We also determine all isomorphic classes of finite commutative rings whose generalized zero divisor graphs have genus at most three.