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.