On rings whose annihilating-ideal graphs are blow-ups of a class of boolean graphs

Abstract

For a finite or an infinite set X, let 2X be the power set of X. A class of simple graph, called strong Boolean graph, is defined on the vertex set 2X \ {X, ∅}, with M adjacent to N if M ⋂ N = ∅. In this paper, we characterize the annihilating-ideal graphs 𝔸𝔾(R) that are blow-ups of strong Boolean graphs, complemented graphs and preatomic graphs respectively. In particular, for a commutative ring R such that 𝔸𝔾(R) has a maximum clique S with 3 ≤ |V (S)| ≤ ∞, we prove that 𝔸𝔾(R) is a blow-up of a strong Boolean graph if and only if it is a complemented graph, if and only if R is a reduced ring. If assume further that R is decomposable, then we prove that 𝔸𝔾(R) is a blow-up of a strong Boolean graph if and only if it is a blow-up of a pre-atomic graph. We also study the clique number and chromatic number of the graph 𝔸𝔾(R).