We know maximal ideals in $C[0,1]$, But what about prime ideals which are not maximal.

This question has already been asked, link

The below answer is by Martin Brandenburg.

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

  1. $\dim(R)=0$
  2. Every prime ideal of $R$ is maximal.
  3. For every $a \in R$ we have $(a^2)=(a)$.
  4. 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).

My question is, Is it really "not possible" to write a non maximal ideal prime ideal explicitly. Pardon me I have misunderstood the answer.

I would love to see a non maximal prime ideal written explicitly!!

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    $\begingroup$ Related: mathoverflow.net/questions/35793 $\endgroup$ – Watson Feb 9 '17 at 13:04
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    $\begingroup$ If $T$ is an existence theorem, then "it is impossible to write down an explicit example of $T$" is very often mathematical shorthand for "$T$ can be false in systems where the axiom of choice fails". So a more precise version of this question would be: are there reasonable (e.g. intuitionistic) axioms besides ZFC that would imply that every prime ideal in $C[0,1]$ is maximal? Every example in the above link appears to use the axiom of choice, so this is a plausible guess. $\endgroup$ – Slade Feb 9 '17 at 13:06

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