Is the function ring $C^{\infty}(M)$ noetherian? Let $M$ be a smooth manifold and $C^{\infty}(M)$ be its function ring. Is this a noetherian ring?

I think that this is not true in general. 
Consider the ring $C^{\infty}(\mathbb{R})$ and the ideal $I$ consisting of smooth functions that vanish on a neighborhood of $0$. I claim that $I$ is not finitely generated. 
Assume by contradiction that $I=(f_{1},\ldots,f_{n})$, where each $f_{i}$ vanishes on a neighborhood $V_{i}$ of $0$. Then each element of $(f_{1},\ldots,f_{n})$ vanishes on $V:=\cap_{i=1}^{n}V_{i}$. Though we can construct smooth functions in $C^{\infty}(\mathbb{R})$ that vanish on an arbitrarily small neighborhood of $0$, in particular strictly smaller than $V$. So $I\neq (f_{1},\ldots,f_{n})$. Hence $I$ is not finitely generated.
