Ring of Infinitely Differentiable Functions Denote the ring $\text{C}^{\infty}$ i.e. infinitely differentiable functions from $\mathbb{R}\mapsto\mathbb{R}$.
I have managed to prove that this ring is a Principal Ideal Domain.
I must also prove that $f(x)=e^{-x^{-2}}$ when $x\neq 0$ and $f(0)=0$ is in $\text{C}^{\infty}$. 
It is clear that $0\in\text{C}^{\infty}$ and for $e^{-x^{-2}}$, I replaced the Taylor Series for $e^x$ and got terms which lie in $\text{C}^\infty$ when $x\neq 0$.
Now I must prove the ring is NOT Noetherian - So consider the set $I_n=\{f(x):f(x)=0, \forall x\geq n\}$. It follows on that the ascending chain of ideals in NOT stationary, and thus the ring is NOT Noetherian.
Finally, I must produce a homomorphism from $\text{C}^\infty\mapsto\mathbb{R}[x]$ using Taylor Series. So,
$$\phi:\text{C}^\infty\mapsto\mathbb{R}[x]$$
$$\phi(f)=\sum_{i=0}^{\infty}\frac{f^{(n)}(0)\ x^n}{n!}$$
For the case of $e^{-x^{-2}}$, we will end up with a Taylor Series of the form
$$e^{-x_0^{-2}}\sum g_i(x)$$
(due to the nature of the exponential function) where $x_0=0$, and thus the Taylor Series is $0$ everywhere (since $f(0)=0$, as defined before). Therefore $e^{-x^{-2}}\in\text{ker}(\phi)$.
Is everything here correct? Or have I made an error somewhere?
 A: The proof that the function
$$
f(x)=\begin{cases}
0 & \text{if $x=0$}\\
\exp(-1/x^2) & \text{if $x\ne0$}
\end{cases}
$$
belongs to $C^{\infty}$ is completely wrong. It's true that $f$ is infinitely differentiable at every point $\ne0$, but the difficult part is exactly showing that it is also infinitely differentiable at $0$. You find a full proof in every textbook (and also on this site, I believe).
Also the fact that $C^{\infty}$ is not noetherian is not sufficiently justified. You have to prove that $I_n$ is a proper subset of $I_{n+1}$; not difficult, though.
You get a homomorphism $C^\infty\to\mathbb{R}[[x]]$ (the ring of formal power series) by considering the Taylor series expansion at $0$.
The image of $f$ (defined above) under this homomorphism is the zero series. Why? When you have completed the proof that $f\in C^{\infty}$ you'll see.
Phrases like “due to the nature of the exponential function” are not allowed in a proof. Besides, if $x_0=0$, $e^{-1/x_0^2}$ means nothing, so you can't use it.
Finally, a principal ideal domain is noetherian. So you can't have possibly proved it is a PID and that it is not noetherian.
