I'm trying to solve the following problem from Atiyah-Macdonald:

Is the ring of continuous function on $[0,1]$ Noetherian ?

Certainly not, here are two non terminating ascending chain of ideals:

$ 1.$ $\langle x^{1/2} \rangle \subset \langle x^{1/4} \rangle \subset \langle x^{1/8} \rangle \subset \langle x^{1/16} \rangle \subset \cdots$, which is clearly ascending and non terminating.

$2.$ Clearly, $I_{ab}=\{f\in R\mid f([a,b])=\{0\}\}$ is an ideal of $C[0,1]$ and for any subinterval $[a,b]$ of $[0,1]$. Notice that if $[c,d]\subseteq [a,b]$, then $I_{cd}\supseteq I_{ab}$. Also, if $[c,d]$ is properly inside $[a,b]$ then the associated ideals are properly contained too because given any closed set, lets say $A$ there always exists a continuous function which exactly vanishes on $A$, hence using this we get a ascending and non terminating chain of ideals of $C[0,1]$

So, my question is:

What are other "interesting" non terminating chains of ideals in $C[0,1]$ or in other words what are all ideals in $C[0,1]$ which are not finitely generated?

As $[0,1]$ is compact, it's well known that all the maximal ideals in $C[0,1]$ are of the following form, $M_a =\{ f\in C[0,1] \mid f(a)=0 \}$ for $a$ $\in$ $[0,1]$, which are not finitely generated. But I don't have any idea of non maximal ideals which are not finitely generated. Any ideas?