The ring of integer-valued polynomials of a Dedekind domain
HTML articles powered by AMS MathViewer
- by Robert Gilmer, William Heinzer, David Lantz and William Smith PDF
- Proc. Amer. Math. Soc. 108 (1990), 673-681 Request permission
Abstract:
Let $D$ be a Dedekind domain and $R = Int(D)$ be the ring of integer-valued polynomials of $D$. We relate the ideal class groups of $D$ and $R$. In particular we prove that, if $D = \mathbb {Z}$ is the ring of rational integers, then the ideal class group of $R$ is a free abelian group on a countably infinite basis.References
- Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
- Demetrios Brizolis, A theorem on ideals in Prüfer rings of integral-valued polynomials, Comm. Algebra 7 (1979), no. 10, 1065–1077. MR 533204, DOI 10.1080/00927877908822391
- Paul-Jean Cahen, Polynomes à valeurs entières, Canadian J. Math. 24 (1972), 747–754 (French). MR 309923, DOI 10.4153/CJM-1972-071-2
- Paul-Jean Cahen and Jean-Luc Chabert, Coefficients et valeurs d’un polynôme, Bull. Sci. Math. (2) 95 (1971), 295–304 (French). MR 296065
- Jean-Luc Chabert, Anneaux de “polynômes à valeurs entières” et anneaux de Fatou, Bull. Soc. Math. France 99 (1971), 273–283 (French). MR 302636
- Jean-Luc Chabert, Un anneau de Prüfer, J. Algebra 107 (1987), no. 1, 1–16 (French, with English summary). MR 883864, DOI 10.1016/0021-8693(87)90068-8
- László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
- Robert Gilmer, Prüfer domains and rings of integer-valued polynomials, J. Algebra 129 (1990), no. 2, 502–517. MR 1040951, DOI 10.1016/0021-8693(90)90233-E
- Robert Gilmer and William W. Smith, Finitely generated ideals of the ring of integer-valued polynomials, J. Algebra 81 (1983), no. 1, 150–164. MR 696131, DOI 10.1016/0021-8693(83)90213-2
- Robert Gilmer and William W. Smith, Integer-valued polynomials and the strong two-generator property, Houston J. Math. 11 (1985), no. 1, 65–74. MR 780821
- Hiroshi Gunji and Donald L. McQuillan, On a class of ideals in an algebraic number field, J. Number Theory 2 (1970), 207–222. MR 257026, DOI 10.1016/0022-314X(70)90021-1
- Irving Kaplansky, Modules over Dedekind rings and valuation rings, Trans. Amer. Math. Soc. 72 (1952), 327–340. MR 46349, DOI 10.1090/S0002-9947-1952-0046349-0
- Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
- Donald L. McQuillan, On Prüfer domains of polynomials, J. Reine Angew. Math. 358 (1985), 162–178. MR 797681, DOI 10.1515/crll.1985.358.162 A. Ostrowski, Über ganzwertige Polynome in algebraischen Zahlkörpern, J. Reine Angew. Math. 149 (1919), 117-124. G. Polya, Über ganzwertige Polynome in algebraischen Zahlkörpern, J. Reine Angew. Math. 149 (1919), 97-116.
- Fusao Shibata, Takasi Sugatani, and Ken-ichi Yoshida, Note on rings of integral-valued polynomials, C. R. Math. Rep. Acad. Sci. Canada 8 (1986), no. 5, 297–301. MR 859430
- H. Zantema, Integer valued polynomials over a number field, Manuscripta Math. 40 (1982), no. 2-3, 155–203. MR 683038, DOI 10.1007/BF01174875
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 108 (1990), 673-681
- MSC: Primary 13F20; Secondary 11R09, 13B25, 13F05
- DOI: https://doi.org/10.1090/S0002-9939-1990-1009989-7
- MathSciNet review: 1009989