Prime ideals in $C[0,1]$ Are there any prime ideals in the ring $C[0,1]$ of continuous functions $[0,1]\rightarrow \mathbb{R}$, which are not maximal?
Perhaps, I duplicate smb's question, but this is an interesting problem!
Could you give me any hint or give a link to some literature?

If $R$ is a reduced commutative ring, then the following statements are equivalent:


*

*$\dim(R)=0$

*Every prime ideal of $R$ is maximal.

*For every $a \in R$ we have $(a^2)=(a)$.

*For every $a \in R$ there is some unit $u \in R$ such that $ua$ is idempotent.


In that case, $R$ is called von Neumann regular. The proof of the equivalences is not so hard. 1. $\Leftrightarrow$ 2. is trivial, 2. $\Rightarrow$ 3. may be reduced to the case of a reduced $0$-dimensional local ring, which has to be a field, for which the claim is obvious, $3. \Rightarrow 2.$ If $\mathfrak{p}$ is a prime ideal, in $R/\mathfrak{p}$ we have $a \equiv a^2 b$ for some $b$, hence $a \equiv 0$ or $1 \equiv ab$, which shows that $R/\mathfrak{p}$ is a field. I leave the equivalence to $4.$ as an exercise.
Applying this to $R=C(K)$ for a perfectly normal space $K$, we see that $\dim(R)=0$ iff $K$ is finite discrete (use that every closed subset of $K$ is the zero set of some $f \in C(K)$, which has to be open-closed by 4.).
In particular, $C[0,1]$ has (lots of) prime ideals which are not maximal. But I don't think that you can write them down explicitly. One can show that every norm-closed prime ideal is maximal (for example using Gelfand duality).
