# When does the complement of the annihilating-ideal graph of a commutative ring admit a cut vertex?

01 Nov 2015-Vol. 2, Iss: 2, pp 9-22

TL;DR: In this article, the authors considered commutative rings with identity that admit at least two nonzero annihilating ideals and a cut vertex in a ring. And they classified rings such that the complement of a ring is connected and admits at least one cut vertex.

Abstract: The rings considered in this article are commutative with identity which admit at least two nonzero annihilating ideals. Let $R$ be a ring. Let $mathbb{A}(R)$ denote the set of all annihilating ideals of $R$ and let $mathbb{A}(R)^{*} = mathbb{A}(R)backslash {(0)}$. The annihilating-ideal graph of $R$, denoted by $mathbb{AG}(R)$ is an undirected simple graph whose vertex set is $mathbb{A}(R)^{*}$ and distinct vertices $I, J$ are joined by an edge in this graph if and only if $IJ = (0)$. The aim of this article is to classify rings $R$ such that $(mathbb{AG}(R))^{c}$ ( that is, the complement of $mathbb{AG}(R)$) is connected and admits a cut vertex.

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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

### "When does the complement of the ann..." refers background in this paper

...Recall from [4] that the zero-divisor graph of R, denoted by Γ(R) is an undirected simple graph whose vertex set is the set of all nonzero zero-divisors of R, and distinct vertices x, y are joined by an edge in this graph if and only if xy = 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

### "When does the complement of the ann..." refers background in this paper

...Beck in [8], several researchers have investigated the interplay between ring theoretic properies of a ring R with the graph theoretic properties...

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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