We denote the rings of all real valued continuous functions on compeletely regular Hausdorff space $X$ by $C(X)$.Let $I$ be a ring ideal of $C(X)$. define $$Z[I]:=\lbrace Z(f):\;f\in I\rbrace$$ where $Z(f):=\lbrace x \in X:\;f(x)=0\rbrace$. We are interested in knowing about the relation about algebraic properties of the ring $C(X)$ and topological properties of the space $X$. I have a question about the relation between the number of zero sets of $I$ in the finite case and it's relation with the topological space $X$. Let me pose my questions in the definite way.
Question1: for which positive integer $n$ we can have a ring ideal $I$, that $Z(I)$ contains only n elements?
Question2: If we have an ideal $I$, such that $|Z(I)|=n$ for some positive integer $n$, can we characterize the ideal $I$?
Question3: for which condition on the space $X$, we have an ideal $I$, with $|Z(I)|=n$, for some positive integer $n$?