C(0, 1) has infinitely many prime ideals not of the form Mp

Question

I know $\mathcal C([0, 1])$ has all maximal ideals of the form $M_p=\{f\in \mathcal C([0, 1]) : f(p) =0,\ p\in [0, 1] \}$. My question is little bit different. If I replace $[0, 1]$ by $(0, 1)$ then how can I show that this ring has infinitely many prime ideals not of the form $M_p$ ?

Answer 1

In the only answer of this thread, Eric Wofsey defines in a more general context $P_k$ which will be a prime ideal in a ring of the form $\mathcal C(X)$. This can be applied to your case, and actually shows that $\mathcal C([0, 1])$ itself has infinitly many prime ideals which are not of the form $M_p$.