Zero-divisor graph of real-valued continuous functions on a frame

Abstract

The main object of this paper is to study the zero-divisor graph Г(RL) of the ring RL. Using the properties of the lattice Coz L, we associate the ring properties of RL, the graph properties of Г(RL), and the properties of a competely regular frame L. Paths in Г(RL) are investigated, and it is shown that the diameter of Г(RL) and the girth of Г(RL) coincide whenever L has at least 5 elements. Cycles in Г(RL) are surveyed, a ring-theoretic and a frame-theoretic characterizations are provided for the graph Г(RL) to be triangulated or be hypertriangulated. We show that Г(RL) is complemented if and only if the space of minimal prime ideals of RL is compact. The relation between the clique number of Г(RL), the cellularity of L and the dominating number of Г(RL) is given. Finally, we prove that if Г(RL) is not triangulated, then the set of centers of Г(RL) is a dominating set if and only if the socle of RL is an essential ideal.