When is CF(X)=MβX∖I(X)CF(X)=MβX∖I(X)?

This short article is in fact an erratum to Theorem 2.5 in Ghirati and Taherifar (2014) [6]. In this paper, we prove that J(C(X)/CF(X))=0J(C(X)/CF(X))=0 or equivalently, the socle CF(X)CF(X) of C(X)C(X) coincides with the intersection of all essential maximal ideals of C(X)C(X) if and only if every infinite subset of I(X)I(X) contains a closed infinite subset, if and only if every pseudocompact subset of X has at most a finite number of isolated points. This fact shows that part (2) of aforementioned Theorem 2.5 is not correct. Our results in this paper also amend Theorem 5.6 in Azarpanah et al. (2008) [2]. Examples are provided to illustrate and delimit our results.