# A new graph associated to a commutative ring

26 May 2016-Discrete Mathematics, Algorithms and Applications (World Scientific Publishing Company)-Vol. 08, Iss: 02, pp 1650029

TL;DR: In this article, the authors consider a simple graph associated with R denoted by ΩR∗, whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I and J are adjacent whenever JAnn(I) = (0) or IAnn(J) =(0).

Abstract: Let R be a commutative ring with identity. In this paper, we consider a simple graph associated with R denoted by ΩR∗, whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I and J are adjacent whenever JAnn(I) = (0) or IAnn(J) = (0). In this paper, we initiate the study of the graph ΩR∗ and we investigate its properties. In particular, we show that ΩR∗ is a connected graph with diam(ΩR∗) ≤ 3 unless R is isomorphic to a direct product of two fields. Moreover, we characterize all commutative rings R with at least two maximal ideals for which ΩR∗ are planar.

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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: The commutative Artinian non-local ring R for which ΩR∗ has genus one and crosscap one is characterized, whose vertex set is the set of all non-trivial ideals of R.

Abstract: Let R be a commutative ring with identity. We consider a simple graph associated with R, denoted by ΩR∗, whose vertex set is the set of all non-trivial ideals of R and two distinct vertices I and J are adjacent whenever JAnn(I) = (0) or IAnn(J) = (0). In this paper, we characterize the commutative Artinian non-local ring R for which ΩR∗ has genus one and crosscap one.

5 citations

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TL;DR: This paper initiates the study of the co-annihilating graph of a commutative ring and investigates its properties.

Abstract: Let R be a commutative ring with identity and 𝔘R be the set of all non-zero non-units of R. The co-annihilating graph of R, denoted by 𝒞𝒜R, is a graph with vertex set 𝔘R and two vertices a and b are adjacent whenever Ann(a) ∩Ann(b) = (0). In this paper, we initiate the study of the co-annihilating graph of a commutative ring and we investigate its properties.

2 citations

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TL;DR: The aim of this paper is to study the interplay between the graph-theoretic properties of [Formula: see text] and the ring-the theoretical properties of [/Formula]: see text.

Abstract: Let R be a commutative ring with identity which is not an integral domain. Let 𝔸(R) denote the set of all annihilating ideals of R and let us denote 𝔸(R)\{(0)} by 𝔸(R)∗. For an ideal I of R, we den...

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TL;DR: This paper characterize all Artinian rings [Formula: see text] for which the genus of [Form formula]: see text is zero or one.

Abstract: Let R be a commutative ring with identity. The co-annihilating-ideal graph of R, denoted by 𝒜R, is a graph whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I ...

1 citations

##### References

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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

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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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17 Dec 1999

TL;DR: In this article, direct-directed graphs are used to describe the properties of trees, independent sets and matchings, and Eulerian and Hamiltonian graphs for graph colouring.

Abstract: Basic Results.- Directed Graphs.- Connectivity.- Trees.- Independent Sets and Matchings.- Eulerian and Hamiltonian Graphs.- Graph Colourings.- Planarity.- Triangulated Graphs.- Applications.

448 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

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TL;DR: In this article, the authors introduced and investigated the total graph of R, denoted by T ( Γ ( R ) ), which is the (undirected) graph with all elements of R as vertices.

290 citations