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\in R\backslash \{۰\} such that Ir = (۰). Let \mathbb{A}(R) denote the set of all annihilating ideals of R and let \mathbb{A}(R)^{*} = \mathbb{A}(R)\backslash \{(۰)\}. Recall that the annihilating-ideal graph of R, denoted by \mathbb{AG}(R), is an undirected graph whose vertex set is \mathbb{A}(R)^{*} and distinct vertices I and J are adjacent in this graph if and only if IJ = (۰). 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.

Keywords:

Annihilating-ideal graph , Clique number , Special principal ideal ring , Zero-dimensional ring