SOME CURIOSITIES OF THE ALGEBRA OF BOUNDED DIRICHLET SERIES Mortini, Raymond; Sasane, Amol;
Abstract
It is shown that the algebra of bounded Dirichlet series is not a coherent ring, and has infinite Bass stable rank. As corollaries of the latter result, it is derived that has infinite topological stable rank and infinite Krull dimension.
E. Amar, Non coherence de certains anneaux de fonctions holomorphes, Illinois J. Math. 25 (1981), no. 1, 68-73.
2.
H. Bass, Algebraic K-Theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968.
3.
H. Bohr, Uber die Bedeutung der Potenzreihen unendlich vieler Variabeln in der Theorie der Dirichletscher Reihen ${\sum}a_n/n^s$, Nachr. Ges. Wiss. Gottingen Math. Phys. 1913 (1913), 441-488.
4.
S. U. Chase, Direct products of modules, Trans. Amer. Math. Soc. 97 (1960), 457-473.
5.
S. Glaz, Commutative coherent rings, Lecture Notes in Mathematics, 1371, Springer-Verlag, Berlin, 1989.
6.
S. Glaz, Commutative coherent rings: historical perspective and current developments, Nieuw Arch. Wisk. (4) 10 (1992), no. 1-2, 37-56.
7.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed. Oxford, at the Clarendon Press, 1954.
8.
H. Hedenmalm, P. Lindqvist, and K. Seip, A Hilbert space of Dirichlet series and systems of dilated functions in $L^2$(0, 1), Duke Math. J. 86 (1997), no. 1, 1-37.
9.
R. C. Heitmann, Generating ideals in Prufer domains, Pacific J. Math. 62 (1976), no. 1, 117-126.
10.
B. Maurizi and H. Queffelec, Some remarks on the algebra of bounded Dirichlet series, J. Fourier Anal. Appl. 16 (2010), no. 5, 676-692.
11.
W. S. McVoy and L. A. Rubel, Coherence of some rings of functions, J. Funct. Anal. 21 (1976), no. 1, 76-87.
12.
R. Mortini, An example of a subalgebra of $H^{\infty}$ on the unit disk whose stable rank is not finite, Studia Math. 103 (1992), no. 3, 275-281.
13.
M. von Renteln, Primideale in der topologischen Algebra $H^{\infty}{\beta}$, Math. Z. 157 (1977), no. 1, 79-82.
14.
M. A. Rieffel, Dimension and stable rank in the K-theory of C*-algebras, Proc. London Math. Soc. (3) 46 (1983), no. 2, 301-333.
15.
K. Seip, Interpolation by Dirichlet series in $H^{\infty}$, In Linear and Complex Analysis, 153-164, Amer. Math. Soc. Transl. Ser. 2, 226, Amer. Math. Soc., Providence, RI, 2009.
16.
D. Suarez, Trivial Gleason parts and the topological stable rank of $H^{\infty}$, Amer. J. Math. 118 (1996), no. 4, 879-904.
17.
S. Treil, The stable rank of the algebra $H^{\infty}$ equals 1, J. Funct. Anal. 109 (1992), no. 1, 130-154.