Computation of depth in C(X)

Abstract

We show that depth(I)≤1 for each ideal I of C(X). This gives a positive answer to a conjecture in Azarpanah et al. (2019) [3]. The present article is in fact an attempt to complete the aforementioned paper. Using the above fact, the depth of some ideals of C(X) such as principal ideals, the ideals Op, p∈βX, and prime ideals are determined in the sense that when they are 0 and when they are 1. We have generalized Proposition 2.9 in that paper and we have shown that depth(C(X)(f)) is at most 1, for each principal ideal (f) in C(X), and it is exactly 1 if and only if intXZ(f) contains at least one non-almost P-point. Also, we prove that for each non-essential principal ideal (f), depth(C(X)(f))=1 or equivalently depth(Ann(f))=1 if and only if the set of non-almost P-points of X is dense in X. Finally, it has been shown that depth(Op)=1, for each p∈βX, if and only if X contains at least two non-almost P-points, and topological spaces X are characterized for which depth(I)=1, for each essential ideal of C(X).