It is clear that each maximal ideal in ring of continuous functions over $[0,1]\subset \mathbb R$ corresponds to a point and vice-versa.

So, for each ideal $I$ define $Z(I) =\{x\in [0,1]\,|\,f(x)=0, \forall f \in I\}$. But map $I\mapsto Z(I)$ from ideals to closed sets is not an injection! (Consider the ideal $J(x_0)=\{f\,|\,f(x)=0, \forall x\in\hbox{ some closed interval which contains }x_0\}$)

How can we describe ideals in $C([0,1])$ ? Is it true that prime ideals are maximal for this ring?

Rings of Continuous Functionsby Gillman & Jerison, already mentioned in Yemon Choi's comment, has sequel of sorts,Rings of Quotients of Rings of Functionsby Fine, Gillman & Lambek, which is also very beautifully written. $\endgroup$ – Gro-Tsen Feb 1 '17 at 17:121more comments