Zero-divisor graph of the rings Cp(x) and Cp∞ (x)

Abstract

In this article we introduce the zero-divisor graphs ?P(X) and ?P? (X) of the two rings CP(X) and CP? (X); here P is an ideal of closed sets in X and CP(X) is the aggregate of those functions in C(X), whose support lie on P. CP? (X) is the P analogue of the ring C?(X). We determine when the weakly zero-divisor graph W?P(X) of CP(X) coincides with ?P(X). We find out conditions on the topology on X, under-which ?P(X) (respectively, ?P? (X)) becomes triangulated/ hypertriangulated. We realize that ?P(X) (respectively, ?P? (X)) is a complemented graph if and only if the space of minimal prime ideals in CP(X) (respectively ?P? (X)) is compact. This places a special case of this result with the choice P ? the ideals of closed sets in X, obtained by Azarpanah and Motamedi in [8] on a wider setting. We also give an example of a non-locally finite graph having finite chromatic number. Finally it is established with some special choices of the ideals P and Q on X and Y respectively that the rings CP(X) and CQ(Y) are isomorphic if and only if ?P(X) and ?Q(Y) are isomorphic.