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?

Consider the sequene of ideals of $C^{\infty}(\mathbb {R}) $, for instance, given by:
$$m \mapsto J_m =\langle \{ e, e^x, e^{x^2}, \cdots e^{x^m}\} \rangle $$
This is an ascending chain of ideals with no upper bound.
The ring is not Noetherian (neither Artinian).
