Zero-Divisor Graphs of Quotient Rings

Journal of Algebra and Related Topics, 2016

For an arbitrary ring $R$, the zero-divisor graph of $R$, denoted by $Gamma (R)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $R$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. It is well-known that for any commutative ring $R$, $Gamma (R) cong Gamma (T(R))$ where $T(R)$ is the (total) quotient ring of $R$. 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.

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).

2008

These rings include chained rings, rings R whose prime ideals contained in Z R are linearly ordered, and rings R such that 0 = Nil R ⊆ zR for all z ∈ Z R \Nil R .

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

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).

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.

Communications in Algebra, 2011

We study the zero divisor graph determined by equivalence classes of zero divisors of a commutative Noetherian ring R. We demonstrate how to recover information about R from this structure. In particular, we determine how to identify associated primes from the graph.

Communications in Algebra, 2006

Let R be a commutative ring with nonzero identity and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by Γ I (R), is the graph whose vertices are the set {x ∈ R \ I| xy ∈ I for some y ∈ R \ I} with distinct vertices x and y adjacent if and only if xy ∈ I. In the case I = 0, Γ 0 (R), denoted by Γ(R), is the zero-divisor graph which has well known results in the literature. In this article we explore the relationship between Γ I (R) ∼ = Γ J (S) and Γ(R/I) ∼ = Γ(S/J). We also discuss when Γ I (R) is bipartite. Finally we give some results on the subgraphs and the parameters of Γ I (R).

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.