ON RINGS WHOSE ANNIHILATING-IDEAL GRAPHS ARE BLOW-UPS OF A CLASS OF BOOLEAN GRAPHS

Jin Guo, Tongsuo Wu, Houyi Yu
2017 Journal of the Korean Mathematical Society  
For a finite or an infinite set X, let 2 X be the power set of X. A class of simple graph, called strong Boolean graph, is defined on the vertex set 2 X \ {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
more » ... G(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).
doi:10.4134/jkms.j160283 fatcat:5lxubkcp4rbqphyxbeeccaez7u