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I know that for a compact manifold $M$ any maximal ideal in the algebra $C^\infty(M)$ of smooth functions on $M$ is of the form $\mathfrak{m}_p=\{f\in C^\infty(M)|f(p)=0\}$. For example, the proof is in the answer to this question.

Is that true for non-compact manifolds? For example, for $M=\mathbb{R}$? I don't think so, but I can't figure out a counterexample. Do you know any example of a maximal ideal in $C^\infty(\mathbb{R})$ which is not of the form $\mathfrak{m}_p$?

Thank you very much!

Edit: Thank you very much for pointing out the maximal ideal, containing the ideal of compactly-supported functions. Does anyone know how to describe it?

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I don't have an explicit example at hand, but there are other ideals than $\mathfrak m_p$. Note that $C_0^\infty$, the set of compactly supported smooth functions is an ideal, not contained in any $\mathfrak m_p$. By Zorn's lemma, every ideal is contained in some maximal ideal, so there must be others.

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  • $\begingroup$ Yes, that's nice! This example didn't come to my mind. $\endgroup$ – Sasha Patotski Dec 17 '13 at 1:21
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No. For example, functions with compact support form an ideal.

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