For any completely regular Hausdorff topological space X, an inter- mediate 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 ZA-ideals in A(X) are examined. It is proved that within the family of almost P-spaces X, each ZA-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). © 2019, Universitat Politecnica de Valencia. All rights reserved.