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

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