A note on zero divisor graph over rings

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.

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.

2018

A zero-divisor graph of a commutative ring R, denoted Γ(R), is a simple graph with vertex set being the set of non-zero zero-divisors of R and with (x, y) an edge if and only if xy = 0. In this paper we study about the zerodivisor graph Γ(R) ,where R is a finite commutative ring. Also we study the compressed zero-divisor graph Γc(R) of R AMS Subject Classification: 05C10, 05C12

Turkish Online Journal of Qualitative Inquiry, 2021

For each commutative ring R we associate a simple graph ⌫ R. We investigate the interplay between the ring-theoretic properties of R and the graph-theo-Ž. retic properties of ⌫ R .

2012

Let R be a commutative ring with nonzero identity, and let Z(R) be its set of zerodivisors. The total graph of R is the (undirected) graph T (Γ(R)) with vertices all elements of R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). In this paper, we study the two (induced) subgraphs Z 0 (Γ(R)) and T 0 (Γ(R)) of T (Γ(R)), with vertices Z(R)\{0} and R\{0}, respectively. We determine when Z 0 (Γ(R)) and T 0 (Γ(R)) are connected and compute their diameter and girth. We also investigate zerodivisor paths and regular paths in T 0 (Γ(R)).

2007

An element a in a ring R is called a strong zero-divisor if, either a b = 0 or b a = 0, for some 0 = b ∈ R (x is the ideal generated by x ∈ R). Let S(R) denote the set of all strong zero-divisors of R. This notion of strong zero-divisor has been extensively studied by these authors in [8]. In this paper, for any ring R, we associate an undirected graph Γ(R) with vertices S(R) * = S(R)\ {0}, where distinct vertices a and b are adjacent if and only if either a b = 0 or b a = 0. We investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of Γ(R). It is shown that for every ring R, every two vertices in Γ(R) are connected by a path of length at most 3, and if Γ(R) contains a cycle, then the length of the shortest cycle in Γ(R), is at most 4. Also we characterize all rings R whose Γ(R) is a complete graph or a star graph. Also, the interplay of between the ring-theoretic properties of a ring R and the graph-theoretic properties of Γ(M n (R)), are fully investigated.

2016

Let r be a positive integer and 2 k ≤ ∈. Let () kr k then it is well known that R is a completely primary finite ring and the structure of its group of units has been studied before. In this paper, we study the structure of its zero divisors via the zero divisor graphs.

2012

Our aim in this note is to study some properties of zero-divisor graphs of Armendariz rings. At first we examine the preservation of completeness of the zero-divisor graph under extension to polynomial and power series rings. Then we study the genus of the certain subrings of upper triangular matrix rings.

Journal of Algebra, 2003

Let Γ(R) be the zero-divisor graph of a commutative ring R. An interesting question was proposed by Anderson, Frazier, Lauve, and Livingston: For which finite commutative rings R is Γ(R) planar? We give an answer to this question. More precisely, we prove that if R is a local ring with at least 33 elements, and Γ(R)≠∅, then Γ(R) is not planar.