Noncoherence of some rings of functions

Sasane, Amol

doi:10.1090/S0002-9939-07-08704-7

Noncoherence of some rings of functions
HTML articles powered by AMS MathViewer

by Amol Sasane PDF

Proc. Amer. Math. Soc. 135 (2007), 2107-2111 Request permission

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

Jacqueline Détraz, Étude du spectre d’algèbres de fonctions analytiques sur le disque unité, C. R. Acad. Sci. Paris Sér. A-B 269 (1969), A833–A835 (French). MR 254604
Sarah Glaz, Commutative coherent rings: historical perspective and current developments, Nieuw Arch. Wisk. (4) 10 (1992), no. 1-2, 37–56. MR 1187898
Kenneth Hoffman, Banach spaces of analytic functions, Dover Publications, Inc., New York, 1988. Reprint of the 1962 original. MR 1102893
Marjan Jerman, When is $C(X)$ a po-coherent ring?, Comm. Algebra 30 (2002), no. 4, 1949–1959. MR 1894053, DOI 10.1081/AGB-120013225
W. S. McVoy and L. A. Rubel, Coherence of some rings of functions, J. Functional Analysis 21 (1976), no. 1, 76–87. MR 0410393, DOI 10.1016/0022-1236(76)90030-6
Charles W. Neville, When is $C(X)$ a coherent ring?, Proc. Amer. Math. Soc. 110 (1990), no. 2, 505–508. MR 943797, DOI 10.1090/S0002-9939-1990-0943797-8
A. Quadrat, The fractional representation approach to synthesis problems: an algebraic analysis viewpoint. I. (Weakly) doubly coprime factorizations, SIAM J. Control Optim. 42 (2003), no. 1, 266–299. MR 1982745, DOI 10.1137/S0363012902417127
A. Quadrat, The fractional representation approach to synthesis problems: an algebraic analysis viewpoint. II. Internal stabilization, SIAM J. Control Optim. 42 (2003), no. 1, 300–320. MR 1982746, DOI 10.1137/S0363012902417139
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
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