# The co-annihilating-ideal graphs of commutative rings

TL;DR: In this article, the co-annihilating ideal graph of a commutative ring with identity was studied and its properties were investigated, where the vertex set is the set of all non-zero proper ideals of and two distinct vertices and are adjacent whenever.

Abstract: Let be a commutative ring with identity. The co-annihilating-ideal graph of , denoted by , is a graph whose vertex set is the set of all non-zero proper ideals of and two distinct vertices and are adjacent whenever . In this paper we initiate the study of the co-annihilating ideal graph of a commutative ring and we investigate its properties.

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TL;DR: In this paper, the strong metric dimension of zero-divisor graphs is studied by transforming the problem of finding the vertex cover number of a strong resolving graph into a more well-known problem.

Abstract: In this paper, we study the strong metric dimension of zero-divisor graph $\Gamma(R)$ associated to a ring $R$. This is done by transforming the problem into a more well-known problem of finding the vertex cover number $\alpha(G)$ of a strong resolving graph $G_{sr}$. We find the strong metric dimension of zero-divisor graphs of the ring $\mathbb{Z}_n$ of integers modulo $n$ and the ring of Gaussian integers $\mathbb{Z}_n[i]$ modulo $n$. We obtain the bounds for strong metric dimension of zero-divisor graphs and we also discuss the strong metric dimension of the Cartesian product of graphs.

9 citations

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TL;DR: The study of the essential ideal graph of a commutative ring with identity was initiated in this article, where the authors investigated its properties and showed that it is a graph whose vertex set is the set of all nonzero proper ideals of R and two vertices I and J are adjacent whenever I + J is an essential ideal.

Abstract: Let R be a commutative ring with identity The essential ideal graph of R, denoted by ℰR, is a graph whose vertex set is the set of all nonzero proper ideals of R and two vertices I and J are adjacent whenever I + J is an essential ideal In this paper, we initiate the study of the essential ideal graph of a commutative ring and we investigate its properties

8 citations

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TL;DR: In this article, the authors consider the problem of classifying rings R such that the complement of the annihilating ideal I of R is planar, i.e., the set of all annihilating ideals of R has a planar complement.

Abstract: The rings considered in this article are commutative with identity which are not integral domains. Let R be a ring. An ideal I of R is said to be an annihilating ideal if there exists $$r\in R\backslash \{0\}$$
such that $$Ir = (0)$$
. Let $$\mathbb {A}(R)$$
denote the set of all annihilating ideals of R and let us denote $$\mathbb {A}(R)\backslash \{(0)\}$$
by $$\mathbb {A}(R)^{*}$$
. Visweswaran and Patel (Discrete Math Algorithms Appl 6:22, 2014) introduced and studied a graph, denoted by $$\Omega (R)$$
, whose vertex set is $$\mathbb {A}(R)^{*}$$
and distinct vertices I, J are joined by an edge in this graph if and only if $$I + J\in \mathbb {A}(R)$$
. In Visweswaran and Sarman (Discrete Math Algorithms Appl 8:22, 2016), we investigated some properties of the complement of $$\Omega (R)$$
. The aim of this article is to classify rings R in order that $$\Omega (R)$$
be planar. We also consider the problem of classifying rings R such that the complement of $$\Omega (R)$$
is planar.

4 citations

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01 Dec 2018TL;DR: In this paper, all commutative Artinian nonlocal rings with genus one have genus one and the annihilating-ideal graph is defined as the graph with the vertices of two distinct vertices that are adjacent if and only if their genus is genus one.

Abstract: Let $R$ be a non-domain commutative ring with identity and $A^*(R)$ be theset of non-zero ideals with non-zero annihilators. We call an ideal $I$ of $R$, anannihilating-ideal if there exists a non-zero ideal $J$ of $R$ such that $IJ = (0)$.The annihilating-ideal graph of $R$ is defined as the graph $AG(R)$ with the vertexset $A^*(R)$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ =(0)$. In this paper, we characterize all commutative Artinian nonlocal rings $R$for which $AG(R)$ has genus one.

3 citations

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01 Dec 2016TL;DR: The study of the sum-annihilating essential ideal graph was initiated in this article, where it was shown that all rings whose essential ideal graphs are complete graphs are stars or complete graphs and sharp bounds on domination number of this graph were established.

Abstract: Let $R$ be a commutative ring with identity. An ideal $I$ of a ring $R$is called an annihilating ideal if there exists $rin Rsetminus {0}$ such that $Ir=(0)$ and an ideal $I$ of$R$ is called an essential ideal if $I$ has non-zero intersectionwith every other non-zero ideal of $R$. Thesum-annihilating essential ideal graph of $R$, denoted by $mathcal{AE}_R$, isa graph whose vertex set is the set of all non-zero annihilating ideals and twovertices $I$ and $J$ are adjacent whenever ${rm Ann}(I)+{rmAnn}(J)$ is an essential ideal. In this paper we initiate thestudy of the sum-annihilating essential ideal graph. We first characterize all rings whose sum-annihilating essential ideal graph are stars or complete graphs and then establish sharp bounds on domination number of this graph. Furthermore determine all isomorphism classes of Artinian rings whose sum-annihilating essential ideal graph has genus zero or one.

3 citations

##### References

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14 Nov 1995

TL;DR: In this article, the authors introduce the concept of graph coloring and propose a graph coloring algorithm based on the Eulers formula for k-chromatic graphs, which can be seen as a special case of the graph coloring problem.

Abstract: 1. Fundamental Concepts. Definitions and examples. Paths and proofs. Vertex degrees and counting. Degrees and algorithmic proof. 2. Trees and Distance. Basic properties. Spanning trees and enumeration. Optimization and trees. Eulerian graphs and digraphs. 3. Matchings and Factors. Matchings in bipartite graphs. Applications and algorithms. Matchings in general graphs. 4. Connectivity and Paths. Cuts and connectivity. k-connected graphs. Network flow problems. 5. Graph Coloring. Vertex colorings and upper bounds. Structure of k-chromatic graphs. Enumerative aspects. 6. Edges and Cycles. Line graphs and edge-coloring. Hamiltonian cycles. Complexity. 7. Planar Graphs. Embeddings and Eulers formula. Characterization of planar graphs. Parameters of planarity. 8. Additional Topics. Perfect graphs. Matroids. Ramsey theory. More extremal problems. Random graphs. Eigenvalues of graphs. Glossary of Terms. Glossary of Notation. References. Author Index. Subject Index.

7,126 citations

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01 Jan 1969

TL;DR: It is shown here how the Noetherian Rings and Dedekind Domains can be transformed into rings and Modules of Fractions using the following structures:

Abstract: * Introduction * Rings and Ideals * Modules * Rings and Modules of Fractions * Primary Decomposition * Integral Dependence and Valuations * Chain Conditions * Noetherian Rings * Artin Rings * Discrete Valuation Rings and Dedekind Domains * Completions * Dimension Theory

4,168 citations

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TL;DR: For each commutative ring R we associate a simple graph Γ(R) as discussed by the authors, and we investigate the interplay between the ring-theoretic properties of R and the graph-theory properties of Γ (R).

1,087 citations

### "The co-annihilating-ideal graphs of..." refers background in this paper

..._e zero divisor graph Γ(R) [8, 9]: _e vertex set of this graph is Z(R) ∖ {0} and two distinct vertices v1 and v2 are adjacent if and only if v1v2 = 0....

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TL;DR: In this article, the authors present the idea of coloring of a commutative ring and show that the existence of an infinite clique implies that the clique R = co implies that there exists an infinitely many cliques.

956 citations

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TL;DR: In this paper, it was shown that if A is a regular Noetherian ring with maximal ideals N 1,..., Ns, such that each A/Ni is finite, then for R = A/Nn11 ··· Nnss, χ(R) = cl(R).

331 citations