Noethérianité de certaines algèbres de fonctions analytiques et applications

Elkhadiri, Abdelhafed; Hlal, Mouttaki

Annales Polonici Mathematici, Tome 75 (2000), p. 247-256 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $M \subset ℝ^{n}$ 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 $S_{K} = f \in H (M) | f (x) \neq 0 f o r a l l x \in K$ ; $S_{K}$ is a multiplicative subset of $H (M)$ . Let $S_{K}^{- 1} H (M)$ be the localization of H(M) with respect to $S_{K}$ . In this paper we prove, first, that $S_{K}^{- 1} H (M)$ is a regular ring (hence noetherian) and use this result in two situations: 1) For each open subset $Ω \subset ℝ^{n}$ , 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, $f_{x}$ , is algebraic on $H (ℝ^{n})$ . We prove that if Ω is a bounded subanalytic subset, then O(Ω) is a regular ring (hence noetherian). 2) Let $M \subset ℝ^{n}$ 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.

Publié le : 2000-01-01

Zbl 0964.32006

EUDML-ID : urn:eudml:doc:208398

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     author = {Elkhadiri, Abdelhafed and Hlal, Mouttaki},
     title = {Noeth\'erianit\'e de certaines alg\`ebres de fonctions analytiques et applications},
     journal = {Annales Polonici Mathematici},
     volume = {75},
     year = {2000},
     pages = {247-256},
     zbl = {0964.32006},
     language = {fra},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv75z3p247bwm}
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Elkhadiri, Abdelhafed; Hlal, Mouttaki. Noethérianité de certaines algèbres de fonctions analytiques et applications. Annales Polonici Mathematici, Tome 75 (2000) pp. 247-256. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv75z3p247bwm/

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