Graphs Associated with Finite Zero Ring (original) (raw)

On zero-divisor graphs of quotient rings and complemented zero-divisor graphs

Journal of Algebra and Related Topics, 2016

For an arbitrary ring RRR, the zero-divisor graph of RRR, denoted by Gamma(R)Gamma (R)Gamma(R), is an undirected simple graph that its vertices are all nonzero zero-divisors of RRR in which any two vertices xxx and yyy are adjacent if and only if either xy=0xy=0xy=0 or yx=0yx=0yx=0. It is well-known that for any commutative ring RRR, Gamma(R)congGamma(T(R))Gamma (R) cong Gamma (T(R))Gamma(R)congGamma(T(R)) where T(R)T(R)T(R) is the (total) quotient ring of RRR. In this paper we extend this fact for certain noncommutative rings, for example, reduced rings, right (left) self-injective rings and one-sided Artinian rings. The necessary and sufficient conditions for two reduced right Goldie rings to have isomorphic zero-divisor graphs is given. Also, we extend some known results about the zero-divisor graphs from the commutative to noncommutative setting: in particular, complemented and uniquely complemented graphs.

On the Zero-Divisor Graph of a Ring

Communications in Algebra, 2008

Let R be a commutative ring with identity, Z(R) its set of zerodivisors, and N il(R) its ideal of nilpotent elements. The zero-divisor graph of R is Γ(R) = Z(R) \ {0}, with distinct vertices x and y adjacent if and only if xy = 0. In this paper, we study Γ(R) for rings R with nonzero zerodivisors which satisfy certain divisibility conditions between elements of R or comparability conditions between ideals or prime ideals of R. These rings include chained rings, rings R whose prime ideals contained in Z(R) are linearly ordered, and rings R such that {0} = N il(R) ⊆ zR for all z ∈ Z(R) \ N il(R).

On the Non-Zero Divisor Graphs of Some Finite Commutative Rings

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.

Some properties of the zero divisor graph of a commutative ring

Discussiones Mathematicae - General Algebra and Applications, 2014

Let Γ(R) be the zero divisor graph for a commutative ring with identity. The k-domination number and the 2-packing number of Γ(R), where R is an Artinian ring, are computed. k-dominating sets and 2-packing sets for the zero divisor graph of the ring of Gaussian integers modulo n, Γ(Z n [i]), are constructed. The center, the median, the core, as well as the automorphism group of Γ(Z n [i]) are determined. Perfect zero divisor graphs Γ(R) are investigated.

A Note on Zero Divisor Graph Over Rings

2007

In this article we discuss the graphs of the sets of zero-divisors of a ring. Now let R be a ring. Let G be a graph with elements of R as vertices such that two non-zero elements a, b ∈ R are adjacent if ab = ba = 0. We examine such a graph and try to find out when

Planar index and outerplanar index of zero-divisor graphs of commutative rings without identity

International Electronic Journal of Algebra

Let RRR be a commutative ring without identity. The zero-divisor graph of R,R,R, denoted by Gamma(R)\Gamma(R)Gamma(R) is a graph with vertex set Z(R)setminus0Z(R)\setminus \{0\}Z(R)setminus0 which is the set of all nonzero zero-divisor elements of R,R,R, and two distinct vertices xxx and yyy are adjacent if and only if xy=0.xy=0.xy=0. In this paper, we characterize the rings whose zero-divisor graphs are ring graphs and outerplanar graphs. Further, we establish the planar index, ring index and outerplanar index of the zero-divisor graphs of finite commutative rings without identity.

The Zero Divisor Graph of the Ring Z_(2^2 p)

ARO-THE SCIENTIFIC JOURNAL OF KOYA UNIVERSITY, 2016

In this paper, we consider the crossing number and the chromatic number of the zero divisor graph Γ() 2 2 Z p to show that this type of zero divisor graphs is bipartite graph, and the smallest cycle in Γ() 2 2 Z p is of length four this implies that the girth is equal four. Index Terms-Bipartite graph, crossing number, girth, planar graph, zero divisor graph of the ring (Z p 2 2).