  We give an example of a non-noetherian quasi-analytic ring constructed using
a quasi-analytic Denjoy-Carleman class. If we denote by $ \mathcal{D}_n$ the
ring of those $ C^\infty$ quasianalytic function germs at $0\in \R^n$ which are
definable in a polynomially bounded o-minimal structure. We show that the
system $\{ \mathcal{D}_n\,/\, n\in\N^*\}$ is not noetherian, i.e. there exists
$m\in\N $, $ m > 1$, such that the ring $\mathcal{D}_m$ is not noetherian.
