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Let $X $ be a Tychonoff topological (completely rgular) space and $C (X) $ be the ring of all real valued functions over $X $. When is the krull dimension of $C (X) $ zero?

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    $\begingroup$ I suppose you mean $C(X)$ is the ring of continuous real-valued functions? Do you know Carl Kohl's 1957 paper "Prime Ideals in Rings of Continuous Functions"? (I didn't really look at it, but it's in my TOREAD list, and the title looks promising.) $\endgroup$
    – Gro-Tsen
    Apr 10, 2017 at 12:51
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    $\begingroup$ In fact, such spaces are called $P$-spaces, and various equivalent conditions are given in Gillman & Jerison's Rings of Continuous Functions (Springer GTM 43), exercise 4J and theorem 14.29 (and notes thereafter). $\endgroup$
    – Gro-Tsen
    Apr 10, 2017 at 13:32

2 Answers 2

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I had written this as a comment, but since the discussion is now a bit confused, it is best to write it as an answer.

The completely regular spaces $X$ such that the ring $C(X)$ is zero-dimensional (i.e., every prime ideal of $C(X)$ is maximal) are known as the "P-spaces" (in the sense of Gillman and Henriksen). The book Rings of Continuous Functions by Gillman and Jerison (Springer 1960, GTM 43) describes a number of properties about them: specifically in exercise 4J and theorem 14.29 (and various other places listed after the latter theorem).

Among the equivalent properties, P-spaces are those in which every function which vanishes at a point $p\in X$ vanishes in a neighborhood of $p$, of in which every $G_\delta$ (countable intersection of open sets) is open.

These spaces look in many ways like discrete spaces, but they are not necessarily discrete: Gillman and Jerison give examples (exercises 4N and 13P) examples of nondiscrete P-spaces.

(Edit.) Here is a simple but interesting example of a non-discrete P-space: consider the set $Q$ of functions $x\colon \omega_1 \to \{0,1\}$ which are eventually $0$ (i.e. there is $\alpha<\omega_1$ such that $x(\xi)=0$ for $\xi\geq\alpha$) and order them lexicographically (i.e., $x$ and $y$ are compared as $x(\xi)$ and $y(\xi)$ for the smallest $\xi$ for which $x(\xi)\neq y(\xi)$). Put the order topoplogy on $Q$. Then $Q$ has no isolated point, but it is still a P-space (Gillman & Jerison, theorem 13.20 + exercise 13.P(1)).

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(Edited, see comments and Gro-Tsen's answer)

Put $A=C(X)$.

(Wrong) I claim that $\dim A=0$ if and only if $X$ is discrete.

The "if" part is easy. Conversely, the condition $\dim A=0$ implies (without any other assumption on $X$) that every $f\in A$ is locally constant. Indeed, fix $f\in A$ and $x\in X$. Denote by $m\subset A$ the corresponding maximal ideal. Then $A_m$ is a reduced zero-dimensional local ring, hence coincides with its residue field $\mathbb{R}$. In particular, the image of $f-f(x)$ in $A_m$ is zero, so $f=f(x)$ in a neighborhood of $x$.

(Wrong) Now if in addition $X$ is a Tychonoff space, every point is the zero set of some $f\in A$, hence must be isolated.

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  • $\begingroup$ The condition that every continuous real-valued function on a space X is locally constant does not imply that the space is discrete, even for Tychonoff spaces. The condition on X that is equivalent to C(X) having Krull dimension 0 is that X be a P-space, and there are certainly non-discrete P-spaces. $\endgroup$
    – Anonymous
    Apr 10, 2017 at 13:16
  • $\begingroup$ @Gro-Tsen: define $f: \mathbb{N} \to [0,1]$ by $f(n) = 1/n$ and extend continuously to $\beta\mathbb{N}$. This function is not locally constant, as it is constantly zero on $\beta\mathbb{N}\setminus\mathbb{N}$, which is not open. $\endgroup$
    – Nik Weaver
    Apr 10, 2017 at 13:54
  • $\begingroup$ @Anonymous: what is a P-space? $\endgroup$ Apr 10, 2017 at 14:11
  • $\begingroup$ @LaurentMoret-Bailly A P-space is precisely what the original question is asking about, i.e., one in which every zero-set is open. My claim that $\beta\mathbb{N}$ is a nondiscrete P-space was wrong, but such spaces do exist: Gillman & Jerison (exercise 4N) give the example of the union of an uncountable discrete set and a point $s$ whose neighborhoods are the co-countable sets containing $s$. $\endgroup$
    – Gro-Tsen
    Apr 10, 2017 at 14:40
  • $\begingroup$ @NikWeaver Ah, thanks, I had misread the statement "if $X$ is a P-space then $\beta X$ is an F-space" in Gilman & Jerison's book (I took the "F" for a "P"). In fact, $\beta X$ is never a P-space if $X$ is infinite because compact P-spaces are indeed discrete. $\endgroup$
    – Gro-Tsen
    Apr 10, 2017 at 14:42

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