Ideals in $C[0,1]$ which are not finitely generated (From Atiyah- Macdonald )

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?

Solution 1:

The vast majority of ideals in $C[0,1]$ are not finitely generated. For instance, if you take a point $p$, and just "randomly" write down some infinite collection of functions which all vanish at $p$, the ideal generated by those functions will very rarely be finitely generated. There is really no hope to classify all possible non-finitely generated ideals.

As a very small taste of the sheer vastness of the different ideals there are in $C[0,1]$, this answer of mine shows that for any point $p\in[0,1]$, there is a chain (in fact, many different chains) of prime ideals which is order-isomorphic to $\mathbb{R}$ and is contained in the ideal of all functions that vanish at $p$.

Some other examples of ideals that may be illuminating: for any closed set $A$ (not just intervals), there is an ideal of functions that vanish on $A$, and this ideal is only finitely generated if $A=[0,1]$ or $A=\emptyset$ (the proof is basically the same as what you did for intervals, except instead of larger intervals you use small neighborhoods of $A$). The set of functions $f$ such that $f(x)/x^n$ approaches $0$ as $x\to 0$ for all $n\in\mathbb{N}$ is an ideal (which is not finitely generated, or even countably generated, though the proof is fairly complicated). For any nonnegative function $f\in C[0,1]$ which vanishes somewhere and does not vanish identically, the ideal $(f,f^{1/2},f^{1/4},f^{1/8},\dots)$ is not finitely generated.