2 LaTeXification.

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?

1

# Maximal ideals in the ring of continuous real-valued functions on R

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=the real line R. Is there a useful characterization of the set of maximal ideals of C(R), the ring of continuous functions on 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 compactification of 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?