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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.
Highlights
The graphs assigned to a commutative ring have been studied by many mathematicians
The zero divisor graph of commutative rings was first introduced by Beck in [1]
We introduce and investigate the notion of maximal ideal graph of a commutative ring R with identity, which is denoted by MG(R)
Summary
The graphs assigned to a commutative ring have been studied by many mathematicians. The zero divisor graph of commutative rings was first introduced by Beck in [1]. We introduce and investigate the notion of maximal ideal graph of a commutative ring R with identity, which is denoted by MG(R). It 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 are maximal ideals of R. The rings R, for which the graph MG(R) is star or complete bipartite, are characterized.
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