Ideal structure of the classical ring of quotients of C(X)

Abstract

In this paper we show that the set of z-ideals and the set of z∘ ideals (=d-ideals) of the classical ring of quotients q(R) (q(X)) of a reduced ring R with property A (C(X)) coincide. Using this fact, we observe that each maximal ideal of q(R) is the extension of a maximal z∘-ideal of R. The members of maximal z∘-ideals of C(X) contained in a given maximal ideal are topologically characterized and using this, it turns out that the extension Op of each Op, p∈βX is a maximal ideal of q(X) if and only if X is a basically disconnected space. Topological spaces X are also characterized for which every Op is contained in a unique maximal ideal of q(X) and in this case, the maximal ideals of q(X) are characterized. Finally, using the concept of z-ideal in q(X), we characterize the regularity of q(X). For instance, we observe that q(X) is regular if and only if for each f∈C(X), there exists a regular (non-zero divisor) element r such that Z(f)∩cozr is open in cozr or equivalently, |f||r| is an idempotent in q(X).