Real-boards of ideals in C(X) and their applications to characterize some spaces

In this paper, using the concept of the real-boards of ideals in C(X), the ring of all real-valued continuous functions on a completely regular Hausdorff space X, some characterizations of nearly pseudocompact and nearly realcompact spaces are given. For an ideal I⊆C(X), the real-board of I is the largest subspace of βX, the Stone-Čech compactification of X, on which every member of I can be extended continuously. Using this concept, the subrings CKI(X) and C∞I(X) of C(X) are introduced and studied, and I-pseudocompact spaces which are generalizations of pseudocompact spaces are presented as well as some of their properties.