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?



Maximal ideals in the ring of continuous real-valued functions on ℝ
Peter Johnstone's book [Stone Spaces](https://books.google.com/books?id=66Njdgsk3ukC&lpg=PR7&pg=PA144#v=onepage&q=&f=false) (p. 144) proves that for any X, maximal ideals in $C(X)$ are the same as maximal ideals in $C\_b(X)$ (bounded functions), i.e. the Stone-Cech compactification $\beta X$. Indeed, if I is a maximal ideal, let Z(I) be the set of all zero sets of elements of I; this is a filter on the lattice of all closed sets that are zero sets of functions. Then I is contained in J(Z(I)), the set of functions whose zero sets are in Z(I), so by maximality they are equal. But also, by maximality, Z(I) must be a maximal filter on the lattice of zero sets, and we get a bijection between maximal filters of zero sets and maximal ideals in $C(X)$. Now the exact same discussion applies to $C\_b(X)$ to give a bijection between maximal filters of zero sets and maximal ideals of $C(X)$ (since the possible zero sets of bounded functions are the same as the possible zero sets of all functions). But the maximal ideals of $C\_b(X)$ are just $\beta X$.


The difference between $C\_b(X)$ and $C(X)$ is that for $C\_b(X)$, the residue fields for all of these maximal ideals are just $C$, while for $C(X)$ you can get more exotic things. Indeed, if a maximal ideal in $C(X)$ has residue field $C$, then every function on X must automatically extend continuously to the corresponding point of $\beta X$. This can actually happen for noncompact X, e.g. the ordinal $\omega\_1$.


Section IV.3 of Johnstone's book has a pretty thorough discussion of this stuff if you want more details.


