A Note on Co-Maximal Ideal Graph of Commutative Rings

الأصل · EN

Let R be a commutative ring with unity. The co-maximal ideal graph of R, denoted by Γ(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 I₁ and I₂ are adjacent if and only if I₁ + I₂ = R. We classify all commutative rings whose co-maximal ideal graphs are planar. In 2012 the following question was posed: If Γ(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.