On the Cozero-Divisor Graphs of Commutative Rings (original) (raw)
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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.
The non-zero divisor graph of a ring
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.
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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.
On the Dot Product Graph of a Commutative Ring
Communications in Algebra, 2014
Let A be a commutative ring with nonzero identity, 1 ≤ n < be an integer, and R = A × A × • • • × A (n times). The total dot product graph of R is the (undirected) graph TD R with vertices R * = R\ 0 0 0 , and two distinct vertices x and y are adjacent if and only if x • y = 0 ∈ A (where x • y denote the normal dot product of x and y). Let Z R denote the set of all zero-divisors of R. Then the zero-divisor dot product graph of R is the induced subgraph ZD R of TD R with vertices Z R * = Z R \ 0 0 0. It follows that each edge (path) of the classical zero-divisor graph R is an edge (path) of ZD R. We observe that if n = 1, then TD R is a disconnected graph and ZD R is identical to the well-known zero-divisor graph of R in the sense of Beck-Anderson-Livingston, and hence it is connected. In this paper, we study both graphs TD R and ZD R. For a commutative ring A and n ≥ 3, we show that TD R (ZD R) is connected with diameter two (at most three) and with girth three. Among other things, for n ≥ 2, we show that ZD R is identical to the zero-divisor graph of R if and only if either n = 2 and A is an integral domain or R is ring-isomorphic to 2 × 2 × 2 .
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
2015
Let A be a commutative ring with nonzero identity, 1 ≤ n < be an integer, and R = A × A × · · · × A (n times). The total dot product graph of R is the (undirected) graph TDDRR with vertices R * = R\0 0 0, and two distinct vertices x and y are adjacent if and only if x · y = 0 ∈ A (where x · y denote the normal dot product of x and y). Let ZZRR denote the set of all zero-divisors of R. Then the zero-divisor dot product graph of R is the induced subgraph ZDDRR of TDDRR with vertices ZZRR * = ZZRR\0 0 0. It follows that each edge (path) of the classical zero-divisor graph RR is an edge (path) of ZDDRR. We observe that if n = 1, then TDDRR is a disconnected graph and ZDDRR is identical to the well-known zero-divisor graph of R in the sense of Beck–Anderson–Livingston, and hence it is connected. In this paper, we study both graphs TDDRR and ZDDRR. For a commutative ring A and n ≥ 3, we show that TDDRR (ZDDRR) is connected with diameter two (at most three) and with girth three. Among other things, for n ≥ 2, we show that ZDDRR is identical to the zero-divisor graph of R if and only if either n = 2 and A is an integral domain or R is ring-isomorphic to 2 × 2 × 2 .
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).
Metric and upper dimension of zero divisor graphs associated to commutative rings
Acta Universitatis Sapientiae, Informatica
Let R be a commutative ring with Z*(R) as the set of non-zero zero divisors. The zero divisor graph of R, denoted by Γ(R), is the graph whose vertex set is Z*(R), where two distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we investigate the metric dimension dim(Γ(R)) and upper dimension dim+(Γ(R)) of zero divisor graphs of commutative rings. For zero divisor graphs Γ(R) associated to finite commutative rings R with unity 1 ≠ 0, we conjecture that dim+(Γ(R)) = dim(Γ(R)), with one exception that R ≅ Π 2 n {\\rm{R}} \\cong \\Pi {\\rm\\mathbb{Z}}_2^{\\rm{n}} , n ≥ 4. We prove that this conjecture is true for several classes of rings. We also provide combinatorial formulae for computing the metric and upper dimension of zero divisor graphs of certain classes of commutative rings besides giving bounds for the upper dimension of zero divisor graphs of rings.