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arXiv · 1307.5401

A Note on Co-Maximal Ideal Graph of Commutative Rings

Saieed Akbari·Babak Miraftab·Reza Nikandish
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Abstract

Let $R$ be a commutative ring with unity. The co-maximal ideal graph of $R$, denoted by $\Gamma(R)$, is a graph whose vertices are the proper ideals of $R$ which are not contained in the Jacobson radical of $R$, and two vertices $I_1$ and $I_2$ are adjacent if and only if $I_1 + I_2 = R$. We classify all commutative rings whose co-maximal ideal graphs are planar. In 2012 the following question was posed: If $\Gamma(R)$ is an infinite star graph, can $R$ be isomorphic to the direct product of a field and a local ring? In this paper, we give an affirmative answer to this question.

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Saieed Akbari, Babak Miraftab, Reza Nikandish. 2013-07-20. A Note on Co-Maximal Ideal Graph of Commutative Rings. https://arxiv.org/abs/1307.5401

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