Here, all functions are real-valued. Let $R = C^\infty((-1, 1))$ be the ring of $C^\infty$ functions on the interval $(-1, 1) \subset \mathbb{R}$. Let $P \subset R$ be a prime ideal defined by $$ P = \{ f \in R \mid (\forall n \in \mathbb{N})(f^{(n)}(0) = 0) \}. $$ I know that $P$ is actually prime.
I am searching for a prime ideal of $R$ that is properly contained in $P$. I guess that the following $P'$ is also a prime ideal of $R$. $$ P' = \{ f \in R \mid (\exists r > 0)(F_r(f)(x) \in P) \}, $$ where we define $$ F_r(f)(x) = \begin{cases} f(x)e^{r/x^2}, & x \neq 0, \\\\ 0, & x = 0. \end{cases} $$ But I don't come up with how to show that $P'$ is prime. Could you tell me how to prove or disprove this? If this is not correct, it will be helpful if you could modify $P'$ to make it prime, thanks.