A class of ideals in intermediate rings of continuous functions

摘要

For? any? completely? regular? Hausdorff? topological? space X,? an? intermediate? ring A(X) of? continuous? functions? stands? for? any? ring? lying between C ? (X) and C(X).? It is a rather recently established fact that if A(X) ≠ C(X), then there exist non maximal prime ideals in A(X).We offer an alternative proof of it on using the notion of z?-ideals.? It is realized that a P-space X is discrete if and only if C(X) is identical to the ring of real valued measurable functions defined on the σ-algebra β(X) of all Borel sets in X.? Interrelation between z-ideals, z?-ideal and ? A -ideals in A(X) are examined.? It is proved that within the family of almost P-spaces X, each ? A -ideal in A(X) is a z?-ideal if and only if each z-ideal in A(X) is a z?-ideal if and only if A(X) = C(X).