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?



prime ideals in C([0,1])
Here is a way to construct a non-maximal prime ideal: consider the multiplicative set $S$
of all non-zero polynomials in $C[0,1]$. Use Zorn lemma to get an ideal $P$ that is disjoint from $S$ and is maximal with this property. $P$ is clearly prime (for this you only need $S$ to be multiplicative.) On the other hand $P$ cannot be any one of the maximal ideals, since
it does not contain $x-c$ for every $c \in [0,1]$.


