On the complement of a graph associated with the set of all nonzero annihilating ideals of a commutative ring

Abstract

The rings considered in this paper are commutative with identity which are not integral domains. Recall that an ideal [Formula: see text] of a ring [Formula: see text] is called an annihilating ideal if there exists [Formula: see text] such that [Formula: see text]. As in [M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10(4) (2011) 727–739], for any ring [Formula: see text], we denote by [Formula: see text] the set of all annihilating ideals of [Formula: see text] and by [Formula: see text] the set of all nonzero annihilating ideals of [Formula: see text]. Let [Formula: see text] be a ring. In [S. Visweswaran and H. D. Patel, A graph associated with the set of all nonzero annihilating ideals of a commutative ring, Discrete Math. Algorithm Appl. 6(4) (2014), Article ID: 1450047, 22pp], we introduced and studied the properties of a graph, denoted by [Formula: see text], which is an undirected simple graph whose vertex set is [Formula: see text] and distinct elements [Formula: see text] are joined by an edge in this graph if and only if [Formula: see text]. The aim of this paper is to study the interplay between the ring theoretic properties of a ring [Formula: see text] and the graph theoretic properties of [Formula: see text], where [Formula: see text] is the complement of [Formula: see text]. In this paper, we first determine when [Formula: see text] is connected and also determine its diameter when it is connected. We next discuss the girth of [Formula: see text] and study regarding the cliques of [Formula: see text]. Moreover, it is shown that [Formula: see text] is complemented if and only if [Formula: see text] is reduced.