The Noetherian property in rings of integer-valued polynomials
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- by Robert Gilmer, William Heinzer and David Lantz PDF
- Trans. Amer. Math. Soc. 338 (1993), 187-199 Request permission
Abstract:
Let $D$ be a Noetherian domain, $D\prime$ its integral closure, and $\operatorname {Int}(D)$ its ring of integer-valued polynomials in a single variable. It is shown that, if $D\prime$ has a maximal ideal $M\prime$ of height one for which $D\prime /M\prime$ is a finite field, then $\operatorname {Int}(D)$ is not Noetherian; indeed, if $M\prime$ is the only maximal ideal of $D\prime$ lying over $M\prime \cap D$, then not even $\operatorname {Spec}(\operatorname {Int}(D))$ is Noetherian. On the other hand, if every height-one maximal ideal of $D\prime$ has infinite residue field, then a sufficient condition for $\operatorname {Int}(D)$ to be Noetherian is that the global transform of $D$ is a finitely generated $D$-module.References
- 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, Dimension de l’anneau des polynômes à valeurs entières, Manuscripta Math. 67 (1990), no. 3, 333–343 (French, with English summary). MR 1046992, DOI 10.1007/BF02568436 —, Polynômes á valeurs entières sur un anneau non analytiquement irreductible, preprint.
- Paul-Jean Cahen and Jean-Luc Chabert, Coefficients et valeurs d’un polynôme, Bull. Sci. Math. (2) 95 (1971), 295–304 (French). MR 296065
- Paul-Jean Cahen, Fulvio Grazzini, and Youssef Haouat, Intégrité du complété et théorème de Stone-Weierstrass, Ann. Sci. Univ. Clermont-Ferrand II Math. 21 (1982), 47–58 (French, with English summary). MR 706121
- Jean-Luc Chabert, Les idéaux premiers de l’anneau des polynômes à valeurs entières, J. Reine Angew. Math. 293(294) (1977), 275–283. MR 441954, DOI 10.1515/crll.1977.293-294.275
- 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 —, Le théorème de Stone-Weierstrass et les polynômes à valeurs entières, Actes du colloque d’algèbre de Montpellier, 1988, (to appear).
- Daniel Ferrand and Michel Raynaud, Fibres formelles d’un anneau local noethérien, Ann. Sci. École Norm. Sup. (4) 3 (1970), 295–311 (French). MR 272779
- 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, William Heinzer, David Lantz, and William Smith, The ring of integer-valued polynomials of a Dedekind domain, Proc. Amer. Math. Soc. 108 (1990), no. 3, 673–681. MR 1009989, DOI 10.1090/S0002-9939-1990-1009989-7
- William Heinzer, On Krull overrings of a Noetherian domain, Proc. Amer. Math. Soc. 22 (1969), 217–222. MR 254022, DOI 10.1090/S0002-9939-1969-0254022-7
- William Heinzer and David Lantz, When is an N-ring Noetherian?, J. Pure Appl. Algebra 39 (1986), no. 1-2, 125–139. MR 816894, DOI 10.1016/0022-4049(86)90140-4
- Jacob R. Matijevic, Maximal ideal transforms of Noetherian rings, Proc. Amer. Math. Soc. 54 (1976), 49–52. MR 387269, DOI 10.1090/S0002-9939-1976-0387269-3
- Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344 —, Commutative ring theory, Cambridge Univ. Press, Cambridge, 1986.
- Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
- Jun-ichi Nishimura, On ideal transforms of Noetherian rings, J. Math. Kyoto Univ. 19 (1979), no. 1, 41–46. MR 527394, DOI 10.1215/kjm/1250522467 A. Ostrowski, Über ganzwertige Polynome in algebraische Zahlkörpern, J. Reine Angew. Math. 358 (1919), 117-124. G. Pólya, Über ganzwertige Polynome in algebraische Zahlkörpern, J. Reine Math. 358 (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
- Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, Graduate Texts in Mathematics, Vol. 29, Springer-Verlag, New York-Heidelberg, 1975. Reprint of the 1960 edition. MR 0389876
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 338 (1993), 187-199
- MSC: Primary 13G05; Secondary 13B22, 13B25, 13E05
- DOI: https://doi.org/10.1090/S0002-9947-1993-1097166-0
- MathSciNet review: 1097166