For a compact space K, $K$, the maximal ideals in the ring C(K) $C(K)$ of continuous real-valued functions on K $K$ are easily identified with the points of K $K$ (a point defines the maximal ideal of functions vanishing at that point).
Now take K=the real line R. $K=\mathbb{R}$. Is there a useful characterization of the set of maximal ideals of C(R), $C(\mathbb{R})$, the ring of continuous functions on R? $\mathbb{R}$? Note that I'm not imposing any boundedness conditions at infinity (if one does, I think the answer has to do with the Stone-Cech Stone-Čech compactification of R $\mathbb{R}$ - 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?
