Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Link between Noetherianity and the Weierstrass Division Theorem on some quasianalytic local rings


Author: Abdelhafed Elkhadiri
Journal: Proc. Amer. Math. Soc. 140 (2012), 3883-3892
MSC (2010): Primary 26E10, 13F25, 32B05, 32B20; Secondary 03C10
DOI: https://doi.org/10.1090/S0002-9939-2012-11243-2
Published electronically: March 20, 2012
MathSciNet review: 2944729
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In the setting of well-behaved quasianalytic differentiable systems, we prove that the Weierstrass Division Theorem holds in such system if, and only if, the system is Noetherian.


References [Enhancements On Off] (What's this?)

  • [1] S.S. Abhyankar and M. van der Put. Homomorphisms of analytic local rings. J. Reine Angew. Math. 242 (1970), 26-60. MR 0260729 (41:5353)
  • [2] C. Andradas, L. Bröcker and J.M. Ruiz. Constructible sets in real geometry. Ergeb. Math., Volume 33, Springer (1996). MR 1393194 (98e:14056)
  • [3] J. Bochnak, M. Coste, and M.-F. Roy. Géométrie algébrique réelle. Springer-Verlag, Berlin and Heidelberg (1987). MR 949442 (90b:14030)
  • [4] C.L. Childress, Weierstrass division in quasianalytic local rings. Can. J. Math., Vol. XXVIII, No. 5, 1976, pp. 938-953. MR 0417441 (54:5491)
  • [5] P.M. Eakin and G.A. Harris, When $ F(f)$ convergent implies $ f$ is convergent, Math. Ann. 229, 201-210 (1977). MR 0444651 (56:3001)
  • [6] A. Elkhadiri and H. Sfouli, Weierstrass division theorem in quasianalytic local rings, Studia Mathematica 185 (1) (2008). MR 2380000
  • [7] A. Elkhadiri, Homomorphism of quasianalytic local rings, Proc. Am. Math. Soc. 138, No. 4, 1433-1438 (2010). MR 2578536 (2011a:32008)
  • [8] C. Miller, Infinite differentiability in polynomially bounded o-minimal structures. Proc. Amer. Math. Soc. 123 (1995), 2551-2555. MR 1257118 (95j:03069)
  • [9] W.F. Osgood, On functions of several complex variables. Trans. Amer. Math. Soc. 17 (1916), 1-8. MR 1501027
  • [10] W. Rudin, Real and complex analysis. McGraw-Hill, New York (1966). MR 0210528 (35:1420)
  • [11] J.-P. Serre, Algèbre locale. Multiplicités, Lecture Notes in Mathematics, 11, Springer-Verlag, Berlin-New York (1965). MR 0201468 (34:1352)
  • [12] B. Teissier, Résultat récents sur l'approximation des morphismes en algèbre commutative. Séminaire Bourbaki, 46ème année, 1993-94, Astérisque 227 (1995), no. 784. MR 1321650 (96c:13023)
  • [13] J.-Cl. Tougeron, Idéaux des fonctions différentiables. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71, Springer-Verlag (1972). MR 0440598 (55:13472)
  • [14] L. Van Den Dries, Tame topology and o-minimal structures. London Mathematical Society, Lecture Note Series, 248. Cambridge University Press, Cambridge (1998). MR 1633348 (99j:03001)
  • [15] O. Zariski and P. Samuel, Commutative Algebra, Volume II, D. Van Nostrand (1960). MR 0120249 (22:11006)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 26E10, 13F25, 32B05, 32B20, 03C10

Retrieve articles in all journals with MSC (2010): 26E10, 13F25, 32B05, 32B20, 03C10


Additional Information

Abdelhafed Elkhadiri
Affiliation: Department of Mathematics, Faculty of Sciences, University Ibn Tofail, B.P. 133, Kénitra, Morocco
Email: kabdelhafed@hotmail.com

DOI: https://doi.org/10.1090/S0002-9939-2012-11243-2
Keywords: Quasianalytic rings, Weierstrass Division Theorem, Noetherian rings
Received by editor(s): January 4, 2011
Received by editor(s) in revised form: May 9, 2011
Published electronically: March 20, 2012
Additional Notes: This work was partially supported by PARS MI33
Communicated by: Franc Forstneric
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society