A division's theorem on some class of ${\mathcal{C}}^{\mathrm{\infty }}$-functions

Let ${\mathcal{E}}_n$ be the ring of the germs of ${\mathcal{C}}^{\mathrm{\infty }}$-functions at the origin in ${\mathbb{R}}^n$. It is well known that if $I$ is an ideal of ${\mathcal{E}}_n$, generated by a finite number of germs of analytic functions, then $I$ is closed. In this paper we consider an ideal of ${\mathcal{E}}_n$ generated by a finite number of germs in some class of ${\mathcal{C}}^{\mathrm{\infty }}$-functions that are not analytic in $\grave{a}$, but quasi-analytic and we shall prove that the result holds in this general situation. We remark that the result is not true for a general ideal of finite type of ${\mathcal{E}}_n$.