Let $R$ be the ring of infinitely differentiable real-valued functions on $(-1, 1)$ under pointwise addition and multiplication, and let $$F(x) = \left\{ \begin{array}{lr} e^{-1/x^4} & \text{if } x\neq 0\\ 0 & \text{if } x=0 \end{array} \right.$$
We are given that $F$ is infinitely differentiable and that all its derivatives vanish at $0$. Then $(F)$ is the principal ideal generated by $F$, and let $A=\sqrt{(F)}$ be the radical of $(F)$. Let $M$ be the maximal ideal of all functions that vanish at $0$, and let $P=\cap_{n=1}^\infty M^n$.
I've showed that $P$ consists of the functions all of whose derivatives vanish at $0$, and that $P$ is a prime ideal. From this, it follows that $F\in P$ and that $A\subseteq P$. I've also shown that this containment is proper, since $$F(x) = \left\{ \begin{array}{lr} e^{-1/x^2} & \text{if } x\neq 0\\ 0 & \text{if } x=0 \end{array} \right.$$ is in $P$ and not in $A$.
Now, I need to find a prime ideal containing $(F)$ that's not $P$ or $M$, and also show that it's properly contained in $P$. From the previous part of the problem, I'm assuming that $A$ is in fact the prime ideal I'm looking for, but I'm having trouble proving that $A$ is prime:
Assume $f$ and $g$ are infinitely differentiable functions such that $fg\in A$. Then, since $A=\sqrt{(F)}$, there is a positive integer $n$, and an infinitely differentiable function $h$ such that $f^ng^n=hF$. So now I need to show that some power of $f$ or $g$ is a multiple of $F$, but this is difficult. I tried dividing both sides by $g^n$, which would write $f^n$ as a multiple of $F$ if I can extend the function $\dfrac{h}{g^n}$ to not have any holes. Now, since $F$ only vanishes at $0$, it follows that if $g$ vanishes at some nonzero point, so does $h$. I guess we can assume that $g^n$ is not a multiple of $F$, though I'm not sure how to use that.
So if $\dfrac{h}{g^n}$ vanishes at $a$, i want to extend this function to $a$ by defining it to be $\displaystyle\lim_{x\to a} \dfrac{h(x)}{g^n(x)}$. Since both numerator and denominator vanish, I can use L'Hopital's Rule to write it as $\dfrac{h'(x)}{ng'(x)g^{n-1}(x)}$. Presumably, I can keep doing this, since the quotient shouldn't go off to infinity, and take $n$ derivatives, whence the numerator becomes $h^{(n)}(x)$, and the denominator becomes the sum of terms which all have a $g(x)$ factor, except for one, which is $g'(x)^n$. If this doesn't vanish, I can get a limit, and if it does, I guess I can do this again for $g''$, etc. The only problematic case is if all of $g$'s derivatives at $a$ vanish. This also means that $h$'s derivatives all vanish at $a$, but I don't think this gives a contradiction, since neither $g$ not $h$ was required to even be in $P$, and even if either of them were, it would be possible for a function's derivatives to all vanish at two different points, right?