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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 ) / C F ( X ) ) = 0 or equivalently, the socle C F ( X ) of C ( X ) coincides with the intersection of all essential maximal ideals of C ( X ) if and only if every infinite subset of 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. [ABSTRACT FROM AUTHOR]

The submodules with the property of the title (N ⊆ M is strongly essential in M if ∏I N is essential in ∏IM for any index set I) are introduced and fully investigated. It is shown that for each submodule N of M there exists a subset T ⊆ M such that N + T is strongly essential submodule of M and (N:T) = Ann(T), T ∩ N = 0. Basic properties of these objects and several examples are given and the counterparts of the related concepts to essential submodules are also introduced and studied. It is shown that each maximal left ideal of a left fully bounded ring is either a summand or strongly essential. Rings over which no module has a proper strongly essential submodule are characterized. It is also shown that the left Loewy rings are the only rings over which the essential submodules and strongly essential submodules of any left module coincide. Finally, a new characterization of left FBN rings is observed. [ABSTRACT FROM AUTHOR]