Zero Divisor Graph of a Commutative Ring

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.

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.

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.

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.

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.

Journal of Algebra, 1999

Ž . Journal of Algebra 217, 434447 1999 Article ID jabr.1998.7840, available online at http:rrwww.idealibrary.com on ... The Zero-Divisor Graph of a Commutative Ring ... David F. Anderson and Philip S. Livingston ... Mathematics Department, The Uni¨ersity of ...

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)).

2014

Let R be a (commutative) ring with nonzero identity and Z.R/ be the set of all zero divisors of R. The total graph of R is the simple undirected graph T. .R// with vertices all elements of R, and two distinct vertices x and y are adjacent if and only if x C y 2 Z.R/. This type of graphs has been studied by many authors. In this paper, we state many of the main results on the total graph of a ring and its related graphs.

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.

Communications in Algebra, 2013

Let R be a commutative ring with nonzero identity, Z R be its set of zero-divisors, and if a ∈ Z R , then let ann R a = d ∈ R da = 0. The annihilator graph of R is the (undirected) graph AG R with vertices Z R * = Z R \ 0 , and two distinct vertices x and y are adjacent if and only if ann R xy = ann R x ∪ ann R y. It follows that each edge (path) of the zero-divisor graph R is an edge (path) of AG R. In this article, we study the graph AG R. For a commutative ring R, we show that AG R is connected with diameter at most two and with girth at most four provided that AG R has a cycle. Among other things, for a reduced commutative ring R, we show that the annihilator graph AG R is identical to the zero-divisor graph R if and only if R has exactly two minimal prime ideals.