A Note on Co-Maximal Ideal Graph of Commutative Rings

Let $R$ be a commutative ring with unity. The co-maximal ideal graph of $R$, denoted by $\mathrm{\Gamma }(R)$, 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 $\mathrm{\Gamma }(R)$ 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.