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

 

Noncoherence of some rings of functions


Author: Amol Sasane
Journal: Proc. Amer. Math. Soc. 135 (2007), 2107-2111
MSC (2000): Primary 46J15, 30A98; Secondary 93C05, 13E15
Published electronically: February 6, 2007
MathSciNet review: 2299487
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Abstract: Let $ \mathbb{D}$, $ \mathbb{T}$ denote the unit disc and unit circle, respectively, in $ \mathbb{C}$, with center 0. If $ S\subset \mathbb{T}$, then let $ A_{S}$ denote the set of complex-valued functions defined on $ \mathbb{D}\cup S$ that are analytic in $ \mathbb{D}$, and continuous and bounded on $ \mathbb{D}\cup S$. Then $ A_{S}$ is a ring with pointwise addition and multiplication. We prove that if the intersection of $ S$ with the set of limit points of $ S$ is not empty, then the ring $ A_{S}$ is not coherent.


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Additional Information

Amol Sasane
Affiliation: Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, United Kingdom
Email: A.J.Sasane@lse.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08704-7
PII: S 0002-9939(07)08704-7
Keywords: Banach algebras of analytic functions, coherent rings
Received by editor(s): September 20, 2005
Received by editor(s) in revised form: March 10, 2006
Published electronically: February 6, 2007
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.