On closed ideals in smooth classes

We study closedness properties of ideals generated by real - analytic funct ions in some subrings C of C-infinity(Omega), where Omega is an open subset of R-n. In contrast with the case C = (CO)-O-infinity(Omega), which has be en clarified by famous works of HORMANDER, LOJASIEWICZ and MALGRANGE, it tu rns out that such ideals are generally not closed when C is an ultradiffere ntiable class. If C is sufficiently regular and non-quasianalytic, and unde r the assumption that the real zero locus of the ideal reduces to a single point, we obtain a sharp sufficient condition of closedness, expressed in t erms of the geometry of common complex zeros for the germs of the generator s at this point. This condition is shown to be also necessary in dimension 2, when the ideal is principal. Some related questions about rings of ultra differentiable germs and about ultradistributions are discussed.