Pasting topological spaces at one point

Let {X _α } _α∈Λ be a family of topological spaces and x _α ∈ X _α , for every α ∈ Λ. Suppose X is the quotient space of the disjoint union of X _α ’s by identifying x _α ’s as one point σ. We try to characterize ideals of C(X) according to the same ideals of C(X _α )’s. In addition we generalize the concept of rank of a point, see 9], and then answer the following two algebraic questions. Let m be an infinite cardinal. (1) Is there any ring R and I an ideal in R such that I is an irreducible intersection of m prime ideals? (2) Is there any set of prime ideals of cardinality m in a ring R such that the intersection of these prime ideals can not be obtained as an intersection of fewer than m prime ideals in R? Finally, we answer an open question in 11].