# Maximal Ideal Graph of Commutative Rings

TL;DR: The notion of maximal ideal graph of a commutative ring with identity was introduced and studied in this article, where the authors introduced the notion of MG(R) and studied its properties and characterizations.

Abstract: In this paper, we introduce and study the notion of the maximal ideal graph of a commutative ring with identity. Let R be a commutative ring with identity. The maximal ideal graph of R, denoted by MG(R), is the undirected graph with vertex set, the set of non-trivial ideals of R, where two vertices I1 and I2 are adjacent if I1 I2 and I1+I2 is a maximal ideal of R. We explore some of the properties and characterizations of the graph.

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TL;DR: In this paper , a new class of rings called involution t-clean ring is introduced which every element in the ring are sum of involution and tripotent elements. And the graph of the ring has diameter one and girth three.

Abstract: A new class of rings are introduced which every element in the ring are sum of involution and tripotent elements. This class called involution t-clean ring which is a generalization of invo-clean rings and subclass of clean rings. Some properties of this class are investigate. For an application in graph theory, new graph is defined called t-clean graph of involution t-clean ring the set of vertices is order pairs of involution and tripotent element which is the sum of them is involution t-clean element. The two vertices are adjacent if and only if the sum of involution elements are zero or the product of the tripotent elements are zero. The graphs are connecting, has diameter one and girth three.

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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: 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, the authors studied the metric dimension of zero-divisor graphs associated with commutative rings and showed that the dimension of these graphs scales with the number of vertices.

Abstract: Let R be a commutative ring with unity 1 and let G(V,E) be a simple graph. In this research article, we study the metric dimension in zero-divisor graphs associated with commutative rings. We show ...

34 citations

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TL;DR: In this article, the annihiator-ideal graph of R is defined as an undirected graph with vertex set (A^*(R)) and two distinct vertices I and J are adjacent if and only if

Abstract: Let R be a commutative ring with nonzero identity. We denote by AG(R) the annihilator graph of R, whose vertex set consists of the set of nonzero zero-divisors of R, and two distint vertices x and y are adjacent if and only if \(\mathrm {ann}(x) \cup \mathrm {ann}(y)
e \mathrm {ann}(xy)\), where for \(t\in R\), we set \(ann(t) := \lbrace r\in R\ \vert \ rt=0\rbrace \). In this paper, we define the annihiator-ideal graph of R, which is denoted by \(A_{I}(R)\), as an undirected graph with vertex set \(A^*(R)\), and two distinct vertices I and J are adjacent if and only if \(\mathrm {ann}(I) \cup \mathrm {ann}(J)
e \mathrm {ann}(IJ)\). We study some basic properties of \(A_{I}(R)\) such as connectivity, diameter and girth. Also we investigate the situations under which the graphs AG(R) and \(A_{I}(R)\) are coincide. Moreover, we examin the planarity of the graph \(A_{I}(R)\).

6 citations

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5 citations