TITLE: Maximal ideals in the ring of continuous real-valued functions on ℝ
QUESTION [25 upvotes]: For a compact space $K$, the maximal ideals in the ring $C(K)$ of continuous real-valued functions on $K$ are easily identified with the points of $K$ (a point defines the maximal ideal of functions vanishing at that point).
Now take $K=\mathbb{R}$. Is there a useful characterization of the set of maximal ideals of $C(\mathbb{R})$, the ring of continuous functions on $\mathbb{R}$? Note that I'm not imposing any boundedness conditions at infinity (if one does, I think the answer has to do with the Stone–Čech compactification of $\mathbb{R}$ — but I can't say I'm totally clear on that part either). Is this ring too large to allow a reasonable description of its maximal ideals?

REPLY [19 votes]: This isn't really an answer to your question, but I'd like to see it here next time I come looking, so I'll post it. The following result is basically Theorem 2.1 in C∞-differentiable spaces by Juan A. Navarro González and Juan B. Sancho de Salas.

Theorem: For any manifold M, the maximal ideals of C(M) whose residue field is ℝ is exactly in bijection with the points of M.

Proof: It's clear that points give you distinct maximal ideals with residue field ℝ, so we just need to show that every such ideal comes from a point. Suppose m is a maximal ideal in C(M) such that C(M)/m=ℝ and  ∩g∈m{g=0}=∅.
Choose a sequence of compact sets K1⊂K2⊂...⊂M such that Ki is in the interior of Ki+1 and M=∪Ki (you can do this since M is hausdorff and second countable). For each i, choose a function fi which is 0 on Ki but 1 outside of Ki+1, and define f=∑fi. Note that for any r∈ℝ, the set {x|f(x)=r} is a closed subset of some Ki, so it is compact.
Since we have a surjection C(M)→ℝ whose kernel is m, there is some r∈ℝ so that f-r∈m. Since ∩g∈m{g=0}=∅, the open sets {g≠0}g∈m is a cover of M, and in particular cover the compact set {f=r}. So there is some finite collection g1, g2, ..., gn∈m so that {g1=0}∩...∩{gn=0}∩{f=r}=∅. But then (g1)²+...+(g1)²+(f-r)²∈m is a nowhere vanishing function, so it is a unit, so m=C(M), a contradiction.