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The Noetherian property in rings of integer-valued polynomials


Authors: Robert Gilmer, William Heinzer and David Lantz
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
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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.


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  • [B] D. Brizolis, A theorem on ideals in Prüfer rings of integral-valued polynomials, Comm. Algebra 7 (1979), 1065-1077. MR 533204 (80j:13013)
  • [Ca1] P.-J. Cahen, Dimension de l'anneau des polynômes a valeurs entières, Manuscripta Math. 67 (1990), 333-343. MR 1046992 (91e:13010)
  • [Ca2] -, Polynômes á valeurs entières sur un anneau non analytiquement irreductible, preprint.
  • [CC] P.-J. Cahen and J.-L. Chabert, Coefficients et valeurs d'un polynôme, Bull. Sci. Math. 95 (1971), 295-304. MR 0296065 (45:5126)
  • [CGH] P.-J. Cahen, F. Grazzini, and Y. Haouat, Intégrité du complété et théorème de Stone-Weierstrass, Ann. Sci. Univ. Clermont-Ferrand II Math. 21 (1982), 47-58. MR 706121 (84j:13005)
  • [Ch1] J.-L. Chabert, Les idéaux premiers de l'anneau des polynômes à valeurs entières. J. Reine Angew. Math. 293/294 (1977), 275-283. MR 0441954 (56:345)
  • [Ch2] -, Un anneau de Prüfer, J. Algebra 107 (1987), 1-16. MR 883864 (88i:13022)
  • [Ch3] -, Le théorème de Stone-Weierstrass et les polynômes à valeurs entières, Actes du colloque d'algèbre de Montpellier, 1988, (to appear).
  • [FR] D. Ferrand and M. Raynaud, Fibres formelles d'un anneau noethérien, Ann. Sci. École Norm. Sup. (4) 3 (1970), 295-311. MR 0272779 (42:7660)
  • [G] R. Gilmer, Prüfer domains and rings of integer-valued polynomials, J. Algebra 129 (1990), 502-517. MR 1040951 (91b:13023)
  • [GHLS] R. Gilmer, W. Heinzer D. Lantz, and W. Smith, The ring of integer-valued polynomials of a Dedekind domain, Proc. Amer. Math. Soc. 108 (1990), 673-681. MR 1009989 (90h:13017)
  • [H] W. Heinzer On Krull overrings of a Noetherian domain, Proc. Amer. Math. Soc. 22 (1969), 217-222. MR 0254022 (40:7235)
  • [HL] W. Heinzer, and D. Lantz, When is an $ N$-ring Noetherian?, J. Pure Appl. Algebra 39 (1986), 125-139. MR 816894 (87b:13023)
  • [Mj] J. Matijevic, Maximal ideal transforms of Noetherian rings, Proc. Amer. Math. Soc.54 (1976), 49-52. MR 0387269 (52:8112)
  • [Mu1] H. Matsumura, Commutative algebra, 2nd ed., Benjamin/Cummings. Reading, Mass., 1980. MR 575344 (82i:13003)
  • [Mu2] -, Commutative ring theory, Cambridge Univ. Press, Cambridge, 1986.
  • [Na] M. Nagata, Local rings, Interscience, New York, London and Sydney, 1962. MR 0155856 (27:5790)
  • [Ni] J. Nishimura, On ideal transforms of noetherian rings. I, J. Math. Kyoto Univ. 19 (1979), 41-46. MR 527394 (80f:13005)
  • [O] A. Ostrowski, Über ganzwertige Polynome in algebraische Zahlkörpern, J. Reine Angew. Math. 358 (1919), 117-124.
  • [P] G. Pólya, Über ganzwertige Polynome in algebraische Zahlkörpern, J. Reine Math. 358 (1919), 97-116.
  • [SSY] F. Shibata, T. Sugatani, and K. Yoshida, Note on rings of integral-valued polynomials, C. R. Math. Rep. Acad. Sci. Canada 8 (1986), 297-301. MR 859430 (87j:13024)
  • [ZS] O. Zariski and P. Samuel, Commutative algebra, vol. I, Springer-Verlag, Berlin, 1975. MR 0389876 (52:10706)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1097166-0
Keywords: Rings of integer-valued polynomials, Noetherian domain, Noetherian spectrum, Prüfer domain, Krull domain, integral closure, global transform
Article copyright: © Copyright 1993 American Mathematical Society

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