characterization of zero-dimensional rings such that the clique number of their annihilating-ideal graphs is at most four

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 of r if there exists r ∈ r{0} such that ir = (0). let a(r) denote the set of all annihilating ideals of r and let a(r) ∗ = a(r){(0)}. recall that the annihilating-ideal graph of r, denoted by ag(r), is an undirected graph whose vertex set is a(r) ∗ and distinct vertices i and j are adjacent in this graph if and only if ij = (0). the aim of this article is to characterize zero-dimensional rings such that the clique number of their annihilating-ideal graphs is at most four.