A class of ideals in intermediate rings of continuous functions
Keywords:P-space, almost P-space, UMP-space, z-ideal, zâ—¦-ideal, ƷA-ideal
For any completely regular Hausdorff topological space X, an intermediate ring A(X) of continuous functions stands for any ring lying between Câˆ—(X) and C(X). It is a rather recently established fact that if A(X) â‰ C(X), then there exist non maximal prime ideals in A(X).We offer an alternative proof of it on using the notion of zâ—¦-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, zâ—¦-ideal and ƷA-ideals in A(X) are examined. It is proved that within the family of almost P-spaces X, each ƷA -ideal in A(X) is a zâ—¦-ideal if and only if each z-ideal in A(X) is a zâ—¦-ideal if and only if A(X) = C(X).
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