Some results on the complement of the comaximal ideal graphs of commutative rings

Abstract

The rings considered in this article are commutative with identity which admit at least two maximal ideals. Let R be a ring. Recall from Ye and Wu (J Algebra Appl 11(6):1250114, 2012) that the comaximal ideal graph of R denoted by ${\mathscr {C}}(R)$ is an undirected graph whose vertex set is the set of all proper ideals I of R such that $I\not \subseteq J(R)$, where J(R) is the Jacobson radical of R and distinct vertices I, J are joined by an edge in this graph if and only if $I + J = R$. The aim of this article is to study the interplay between the ring-theoretic properties of R and the graph-theoretic properties of $({\mathscr {C}}(R))^{c}$, where $({\mathscr {C}}(R))^{c}$ is the complement of the comaximal ideal graph of R.