  It is shown that Denjoy-Carleman quasi analytic rings of germs of functions
in two or more variables either complex or real valued that are stable under
derivation and strictly larger than the ring of real-analytic germs are not
Noetherian rings. The failure of Weierstrass division on these Denjoy-Carleman
classes yields a contradiction to Noetherianity via a stronger version of Artin
Approximation due to Popescu as well as results on projective modules. This
settles a 35-year old open problem in real algebraic geometry.
