Some results on a supergraph of the sum annihilating ideal graph of a commutative ring

The rings considered in this paper are commutative with identity which are not integral domains. Let [Formula: see text] be a ring. An ideal [Formula: see text] of [Formula: see text] is said to be an annihilating ideal if there exists [Formula: see text] such that [Formula: see text]. Let [Formula: see text] denote the set of all annihilating ideals of [Formula: see text] and we denote [Formula: see text] by [Formula: see text]. With [Formula: see text], in this paper, we associate an undirected graph denoted by [Formula: see text] whose vertex set is [Formula: see text] and two distinct vertices [Formula: see text] are adjacent in this graph if and only if either [Formula: see text] or [Formula: see text]. The aim of this paper is to study the interplay between some graph properties of [Formula: see text] and the algebraic properties of [Formula: see text] and to compare some graph properties of [Formula: see text] with the corresponding graph properties of the annihilating ideal graph of [Formula: see text] and the sum annihilating ideal graph of [Formula: see text].