# Noethérianité de certaines algèbres de fonctions analytiques et applications

• Volume: 75, Issue: 3, page 247-256
• ISSN: 0066-2216

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## Abstract

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

## How to cite

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Elkhadiri, Abdelhafed, and Hlal, Mouttaki. "Noethérianité de certaines algèbres de fonctions analytiques et applications." Annales Polonici Mathematici 75.3 (2000): 247-256. <http://eudml.org/doc/208398>.

author = {Elkhadiri, Abdelhafed, Hlal, Mouttaki},
journal = {Annales Polonici Mathematici},
keywords = {Nash functions; regular rings; analytic algebra; subanalytic sets},
language = {fre},
number = {3},
pages = {247-256},
title = {Noethérianité de certaines algèbres de fonctions analytiques et applications},
url = {http://eudml.org/doc/208398},
volume = {75},
year = {2000},
}

TY - JOUR
AU - Elkhadiri, Abdelhafed
AU - Hlal, Mouttaki
TI - Noethérianité de certaines algèbres de fonctions analytiques et applications
JO - Annales Polonici Mathematici
PY - 2000
VL - 75
IS - 3
SP - 247
EP - 256
LA - fre
KW - Nash functions; regular rings; analytic algebra; subanalytic sets
UR - http://eudml.org/doc/208398
ER -

## References

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1. [1] J. Bochnak, M. Coste et M.-F. Roy, Géométrie algébrique réelle, Ergeb. Math. Grenzgeb. 12, Springer, New York, 1987. Zbl0633.14016
2. [2] M. Coste, J. M. Ruiz and M. Shiota, Approximation in compact Nash manifolds, Amer. J. Math. 117 (1995), 905-927. Zbl0873.32007
3. [3] A. Elkhadiri et J.-Cl. Tougeron, Familles noethériennes de modules sur $\underline{k}\left[\left[x\right]\right]$ et applications, Bull. Sci. Math. 120 (1996), 253-292.
4. [4] H. Matsumura, Commutative Algebra, Benjamin, New York, 1970.
5. [5] D. Popescu, General Neron desingularization, Nagoya Math. J. 100 (1985), 97-126. Zbl0561.14008

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