Some results on a spanning subgraph of the complement of the annihilating-ideal graph of a commutative reduced ring

Abstract

Let [Formula: see text] be a commutative ring with identity which is not an integral domain. Let [Formula: see text] denote the set of all annihilating ideals of [Formula: see text] and let us denote [Formula: see text] by [Formula: see text]. For an ideal [Formula: see text] of [Formula: see text], we denote the annihilator of [Formula: see text] in [Formula: see text] by [Formula: see text]. That is, [Formula: see text]. In this note, for any ring [Formula: see text] with [Formula: see text], we associate an undirected graph denoted by [Formula: see text] whose vertex set is [Formula: see text] and distinct vertices [Formula: see text] are joined by an edge if and only if either [Formula: see text] or [Formula: see text]. Let [Formula: see text] be a reduced ring. The aim of this paper is to study the interplay between the graph-theoretic properties of [Formula: see text] and the ring-theoretic properties of [Formula: see text].