Exercise 15, § 2

Let $X$ be a completely regular topological space and $\beta X$ its Stone-Čech compactification (General Topology, Chapter IX, § 1, Exercise 7). For all $x \in X$, let $\mathfrak{g}(x)$ denote the maximal ideal of the ring $\mathscr{C}(X; \mathbf{R})$ consisting of those $f \in \mathscr{C}(X; \mathbf{R})$ such that $x$ belongs to the closure of the sub set $\overline{f}(0)$ of $X$ (General Topology, Chapter X, § 4, Exercise 15); let $\mathfrak{u}(x)$ denote the ideal consisting of those $f \in \mathscr{C}(X; \mathbf{R})$ such that $\overline{f}(0)$ is the trace on $X$ of a neighbourhood of $x$ in $X$; $\mathfrak{u}(x)$ is equal to its radical.

An ideal $a$ of $\mathscr{C}(X; \mathbf{R})$ is called isolated (resp. absolutely isolated) if, for all $f \geq 0$ on $a$, every $g$ such that $0 \leq g \leq f$ (resp. $|g| \leq f$) belongs to $a$ (cf. Algebra, Chapter IV, § 1, Exercise 4); then $\mathscr{C}(X; \mathbf{R})/a$ possesses an order structure (resp. a lattice-order structure) compatible with its ring structure and for which the elements $\geq 0$ are the classes mod. $a$ of the elements $\geq 0$ to $\mathscr{C}(X; \mathbf{R})$ (loc. cit.).

(a) Show that an ideal $a$ of $\mathscr{C}(X; \mathbf{R})$ which is equal to its radical is absolutely isolated (if $f \in a$ and $|g| \leq |f|$, consider the function equal to 0 when $f(x) = 0$ and to $(g(x))^2/f(x)$ when $f(x) \neq 0$). In this case for $\mathscr{C}(X; \mathbf{R})/a$ to be totally ordered, it is necessary and sufficient that $a$ be prime; the set of prime ideals of $\mathscr{C}(X; \mathbf{R})/a$ is then totally ordered by inclusion.

(b) For an isolated ideal $a$ of $\mathscr{C}(X; \mathbf{R})$ to be such that $\mathscr{C}(X; \mathbf{R})/a$ is totally ordered, it is necessary that $a$ be contained in a single maximal ideal $\mathfrak{g}(x)$ and then $a$ necessarily contains $\mathfrak{u}(x)$; it is sufficient that $a$ contain a prime ideal and then $a$ is absolutely isolated (note that the relation $fg = 0$ then implies $f \in a$ or $g \in a$).

(c) Take $X = \mathbf{R}$, $x = 0$; show that the principal ideal generated by the function $t \mapsto |t|$ is contained in $\mathfrak{g}(0)$ and contains $\mathfrak{u}(0)$ but is not isolated; the principal ideal generated by the identity mapping $t \mapsto t$ is contained in $\mathfrak{g}(0)$, contains $\mathfrak{u}(0)$ and is isolated but not absolutely isolated.

(d) Show that, if $p$ and $q$ are two prime ideals in $\mathscr{C}(X; \mathbf{R})$, then $pq = p \cap q$ (and in particular $p^2 = p$) and $p + q$ is prime or equal to $\mathscr{C}(X; \mathbf{R})$. (If $p + q \neq \mathscr{C}(X; \mathbf{R})$, observe that $p + q$ is absolutely isolated and equal to its radical and use (a).)

(e) For $\mathfrak{u}(x)$ to be prime, it is necessary and sufficient that, for every function $f \in \mathscr{C}(X; \mathbf{R})$, the closure in $\beta X$ of one of the two sets $\overline{f^1}([0, +\infty[)$, $\overline{f^1}([-\infty, 0])$ is a neighbourhood of $x$. Deduce an example where $\mathfrak{u}(x)$ is prime and not maximal (consider the topological space associated with a non-trivial ultrafilter; cf. General Topology, Chapter I, § 6, no. 5).

(f) Suppose that $x \in X$. For $\mathscr{U}(x) = \mathfrak{F}(x)$, it is necessary and sufficient that every countable intersection of neighbourhoods of $x$ in $X$ be a neighbourhood of $x$ in $X$. (Cf. General Topology, Chapter I, § 7, Exercise 7.)

(g) Show that $\mathscr{C}(X; \mathbf{R})$ is canonically identified with the total ring of fractions of $\mathscr{C}^\infty(X; \mathbf{R})$.