Noethérianité de certaines algèbres de fonctions analytiques et applications

Let ${\displaystyle M\subset {ℝ}^n}$ be a real-analytic submanifold and H(M) the algebra of real analytic functions on M. If K ⊂ M is a compact subset we consider ${\displaystyle S_K=\{f\in H\left(M\right)\mid f\left(x\right)\ne 0 \text{for all} x\in K\}}$; ${\displaystyle S_K}$ is a multiplicative subset of ${\displaystyle H\left(M\right)}$. Let ${\displaystyle S_K^{-1}H\left(M\right)}$ be the localization of H(M) with respect to ${\displaystyle S_K}$. In this paper we prove, first, that ${\displaystyle S_K^{-1}H\left(M\right)}$ is a regular ring (hence noetherian) and use this result in two situations:    1) For each open subset ${\displaystyle \Omega \subset {ℝ}^n}$, we denote by O(Ω) the subalgebra of H(Ω) defined as follows: f ∈ O(Ω) if and only if for all x ∈ Ω, the germ of f at x, ${\displaystyle f_x}$, is algebraic on ${\displaystyle H\left({ℝ}^n\right)}$. We prove that if Ω is a bounded subanalytic subset, then O(Ω) is a regular ring (hence noetherian).    2) Let ${\displaystyle M\subset {ℝ}^n}$ be a Nash submanifold and N(M) the ring of Nash functions on M; we have an injection N(M) → H(M). In [2] it was proved that every prime ideal p of N(M) generates a prime ideal of analytic functions pH(M) if M or V(p) is compact. We use our Theorem 1 to give another proof in the situation where V(p) is compact. Finally we show that this result holds in some particular situation where M and V(p) are not assumed to be compact.