Let ${\left\{X_{\alpha }\right\}}_{\alpha \in \Lambda }$ be a family of topological spaces and $x_{\alpha }\in X_{\alpha }$, for every $\alpha \in \Lambda$. Suppose $X$ is the quotient space of the disjoint union of $X_{\alpha }$’s by identifying $x_{\alpha }$’s as one point $\sigma$. We try to characterize ideals of $C\left(X\right)$ according to the same ideals of $C\left(X_{\alpha }\right)$’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...