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)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-07-08704-7
Published electronically: February 6, 2007
MathSciNet review: 2299487
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


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

  • 1. J. Détraz. Étude du spectre d'algèbres de fonctions analytiques sur le disque unité. (French) Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B, 269:A833-A835, 1969. MR 0254604 (40:7812)
  • 2. S. Glaz. Commutative coherent rings: historical perspective and current developments. Nieuw Archief voor Wiskunde, nos. 1-2, 4:37-56, 1992. MR 1187898 (93j:13024)
  • 3. K. Hoffmann. Banach Spaces of Analytic Functions. Dover, 1988. MR 1102893 (92d:46066)
  • 4. M. Jerman. When is $ C(X)$ a po-coherent ring? Communications in Algebra, no. 4, 30:1949-1959, 2002. MR 1894053 (2003b:06018)
  • 5. W.S. McVoy and L.A. Rubel. Coherence of some rings of functions. Journal of Functional Analysis, 21:76-87, 1976. MR 0410393 (53:14143)
  • 6. C.W. Neville. When is $ C(X)$ a coherent ring? Proceedings of the American Mathematical Society, no. 2, 110:505-508, 1990.MR 0943797 (90m:54024)
  • 7. A. Quadrat. The fractional representation approach to synthesis problems: an algebraic analysis viewpoint. Part I: (Weakly) doubly coprime factorizations. SIAM Journal on Control and Optimization, 42:266-299, 2004. MR 1982745 (2004f:93062)
  • 8. A. Quadrat. The fractional representation approach to synthesis problems: an algebraic analy- sis viewpoint. Part II: Internal Stabilization. SIAM Journal on Control and Optimization, 42:300-320, 2004. MR 1982746 (2004f:93063)
  • 9. W. Rudin. Real and Complex Analysis. 3rd Edition, McGraw-Hill, 1987.MR 0924157 (88k:00002)
  • 10. A.J. Sasane. Irrational transfer function classes, coprime factorization and stabilization. CDAM Research Report CDAM-LSE-2005-10, May 2005. Available electronically at http://www.cdam.lse.ac.uk/Reports/Files/cdam-2005-10.pdf

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46J15, 30A98, 93C05, 13E15

Retrieve articles in all journals with MSC (2000): 46J15, 30A98, 93C05, 13E15


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: https://doi.org/10.1090/S0002-9939-07-08704-7
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.

American Mathematical Society