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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Orderings and maximal ideals of rings of analytic functions


Author: A. Díaz-Cano
Journal: Proc. Amer. Math. Soc. 133 (2005), 2821-2828
MSC (2000): Primary 14P15, 32B15, 32B20
Published electronically: March 24, 2005
MathSciNet review: 2159758
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Abstract: We prove that there is a natural injective correspondence between the maximal ideals of the ring of analytic functions on a real analytic set $X$ and those of its subring of bounded analytic functions. By describing the maximal ideals in terms of ultrafilters we see that this correspondence is surjective if and only if $X$ is compact. This approach is also useful for studying the orderings of the field of meromorphic functions on $X$.


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Additional Information

A. Díaz-Cano
Affiliation: Departamento de Álgebra, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
Email: Antonio_DiazCano@mat.ucm.es

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07848-2
PII: S 0002-9939(05)07848-2
Keywords: Real analytic sets, analytic functions, maximal ideals, ultrafilters, orderings
Received by editor(s): November 20, 2002
Received by editor(s) in revised form: May 20, 2004
Published electronically: March 24, 2005
Additional Notes: This work was supported by the European Community’s Human Potential Programme under contract HPRN-CT-2001-00271, RAAG and by the Spanish Research Project GAAR BFM2002-04797.
Dedicated: Dedicated to Eberhard Becker on the occasion of his 60th birthday
Communicated by: Michael Stillman
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.