On the planarity of a graph associated to a commutative ring and on the planarity of its complement

The rings considered in this article are commutative with identity which are not integral domains. Let R be a ring. An ideal I of R is said to be an annihilating ideal if there exists r∈ R { 0 } such that Ir= (0 ). Let A(R) denote the set of all annihilating ideals of R and let us denote A(R) { (0 ) } by A(R) ∗. Visweswaran and Patel (Discrete Math Algorithms Appl 6:22, 2014) introduced and studied a graph, denoted by Ω (R) , whose vertex set is A(R) ∗ and distinct vertices I, J are joined by an edge in this graph if and only if I+ J∈ A(R). In Visweswaran and Sarman (Discrete Math Algorithms Appl 8:22, 2016), we investigated some properties of the complement of Ω (R). The aim of this article is to classify rings R in order that Ω (R) be planar. We also consider the problem of classifying rings R such that the complement of Ω (R) is planar. © 2017, Instituto de Matemática e Estatística da Universidade de São Paulo.