Let $R=\dfrac{C^{\infty}(\mathbb{R}^{n})}{I}$ the ring of germs of functions at $0$, where $I$ is the ideal of smooth functions that vanishes in a neighborhood of $0$.
Defining the avaliation homomorphism $a: R \rightarrow \mathbb{R}$ by $$\bar{f} \rightarrow a(f)=f(0),$$ and using the Taylor formula in $0$ follows from Isomorphism theorem that $\ker a=(\bar{X_1}, \ldots, \bar{X_n})$ is the only maximal ideal of $R$.
My problem is
How can I show that $$\bigcap_{k=0}^{\infty}(\ker a)^{k} \ne (0)$$ is not finitely generated?
Thank you in advanced.