Maximal ideal containing functions with compact support I recently proved the following statement:
Let $M$ be a smooth manifold and let $I \subseteq C^\infty(M)$ be an ideal such that $C^\infty(M)/I \cong \mathbb{R}$ (such an ideal is clearly maximal, since $\mathbb{R}$ is a field). Then $I= \mathfrak{m}_p$ for some $p \in M$, where $\mathfrak{m}_p= \left\{f \in C^\infty(M): f(p)=0 \right\}$. 
This is fairly easy to prove for compact $M$. The case for non-compact $M$ requires a bit more work. Now, take $M=\mathbb{R}$ and $I$ the ideal of functions with compact support. $I \nsubseteq \mathfrak{m}_p$ for any $p \in \mathbb{R}$, but there must exist a maximal ideal $I' \subseteq C^\infty(\mathbb{R})$ containing $I$. Clearly $I' \neq \mathfrak{m}_p$, so if $C^\infty(\mathbb{R})/I' \cong \mathbb{F}$ for a field $\mathbb{F}$, which must be the case since $I'$ is maximal, $\mathbb{F} \neq \mathbb{R}$. 
I have two questions: Does anyone have an explicit example of such an $I'$? Also, does anyone know what $\mathbb{F}$ is or could be?
 A: Here is an "explicit" example.  Let $U$ be a nonprincipal ultrafilter on $\mathbb{N}$.  Define $I'$ to be the set of $f\in C^\infty(\mathbb{R})$ such that $\{n\in\mathbb{N}:f(n)=0\}\in U$.  It is easy to see that $I'$ is an ideal, and it contains $I$ since $U$ is nonprincipal.  To show $I'$ is maximal, note that if $f\not\in I'$, then $\{n\in\mathbb{N}:f(n)\neq 0\}\in U$ since $U$ is an ultrafilter.  We can find a smooth function $g$ such that $g(n)=1/f(n)$ for each $n\in\mathbb{N}$ such that $f(n)\neq 0$, and then $fg-1\in I$.  Thus $f$ is a unit mod $I'$, so the quotient $C^\infty(\mathbb{R})/I'$ is a field.
To understand this quotient field $C^\infty(\mathbb{R})/I'$, note that any sequence of real numbers can be realized as the values $f(n)$ for $n\in\mathbb{N}$ for some smooth function $f$.  Moreover, two functions which agree on $\mathbb{N}$ are equal mod $I$.  So in taking the quotient, we can replace smooth functions by just their sequences of values on $\mathbb{N}$, and we find that $C^\infty(\mathbb{R})/I'$ is the ring of sequences $(x_n)$ of real numbers mod the ideal of sequences such that $\{n:x_n=0\}\in U$.  This field is better-known as the ultrapower of $\mathbb{R}$ by the ultrafilter $U$, sometimes written $\prod_U\mathbb{R}$.  This field is also sometimes called a field of "hyperreal numbers" and is studied in nonstandard analysis as a field with very deep similarities to $\mathbb{R}$ but which has infinitesimal elements.
Of course, all of this is only as "explicit" as our choice of nonprincipal ultrafilter $U$ is, and there's not really any way to explicitly write down a nonprincipal ultrafilter.
