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Integer-valued polynomials on a subset. (English) Zbl 0781.13013
Let $$D$$ be an integral domain, which is not a field, $$K$$ its quotient field, $$D'$$ the integral closure of $$D$$ and $$\emptyset\subsetneqq E\subseteqq K$$. Let $$\text{Int}(E,D)=\bigl\{f\in K[X];f(E)\subseteqq D\bigr\}$$ be the ring of integer-valued polynomials. The paper is devoted to the study of the prime ideals of $$\text{Int}(E,D)$$ over the maximal ideal $${\mathfrak m}$$ of $$D$$ in the case where $$D$$ is unibranched, that means, $$D$$ is noetherian, one-dimensional, local and $$D'$$ is local, and $$E$$ is a fractional subset of $$D$$. One of the main results states that under the additional assumption that $$D/{\mathfrak m}$$ is finite these prime ideals are in one-to-one correspondence with the elements of the topological closure $$\overline E$$ of $$E$$ in the completion $$\hat K$$ of $$K$$ for the topology defined by the valuation of $$D'$$: to any element $$\alpha$$ of $$\overline E$$ corresponds the prime ideal $${\mathfrak M}_ \alpha=\bigl\{f\in\text{Int}(E,D);f(\alpha)\in\hat{\mathfrak m}'\bigr\}$$. Other results deal with characterizations of the properties “$$\text{Int}(E,D)$$ is noetherian” and “$$\text{Int}(D,D')=D'[X]$$”, respectively.
Reviewer: G.Kowol (Wien)

##### MSC:
 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 13A18 Valuations and their generalizations for commutative rings 13G05 Integral domains 13E05 Commutative Noetherian rings and modules 13B25 Polynomials over commutative rings
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##### References:
 [1] David F. Anderson, Alain Bouvier, David E. Dobbs, Marco Fontana, and Salah Kabbaj, On Jaffard domains, Exposition. Math. 6 (1988), no. 2, 145 – 175. · Zbl 0657.13011 [2] Ahmed Ayache and Paul-Jean Cahen, Anneaux vérifiant absolument l’inégalité ou la formule de la dimension, Boll. Un. Mat. Ital. B (7) 6 (1992), no. 1, 39 – 65 (French, with Italian summary). · Zbl 0785.13001 [3] N. Bourbaki, Algèbre commutative, Hermann, Paris, 1961/1965. · Zbl 0108.04002 [4] Demetrios Brizolis, Ideals in rings of integer valued polynomials, J. Reine Angew. Math. 285 (1976), 28 – 52. · Zbl 0326.13009 [5] Paul-Jean Cahen, Polynomes à valeurs entières, Canad. J. Math. 24 (1972), 747 – 754 (French). · Zbl 0224.13006 [6] Paul-Jean Cahen, Couples d’anneaux partageant un idéal, Arch. Math. (Basel) 51 (1988), no. 6, 505 – 514 (French). · Zbl 0668.13005 [7] Paul-Jean Cahen, Dimension de l’anneau des polynômes à valeurs entières, Manuscripta Math. 67 (1990), no. 3, 333 – 343 (French, with English summary). · Zbl 0712.13001 [8] Paul-Jean Cahen, Polynômes à valeurs entières sur un anneau non analytiquement irréductible, J. Reine Angew. Math. 418 (1991), 131 – 137 (French). · Zbl 0722.13005 [9] -, Parties pleines d’un anneau noethérien, preprint 91-5 de l’URA 225, J. Algebra (to appear). · Zbl 0780.13013 [10] Paul-Jean Cahen and Jean-Luc Chabert, Coefficients et valeurs d’un polynôme, Bull. Sci. Math. (2) 95 (1971), 295 – 304 (French). · Zbl 0221.13006 [11] 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). · Zbl 0527.13015 [12] Jean-Luc Chabert, Anneaux de ”polynômes à valeurs entières” et anneaux de Fatou, Bull. Soc. Math. France 99 (1971), 273 – 283 (French). · Zbl 0202.32802 [13] Jean-Luc Chabert, Les idéaux premiers de l’anneau des polynômes à valeurs entières, J. Reine Angew. Math. 293/294 (1977), 275 – 283. · Zbl 0349.13009 [14] -, Polynômes à valeurs entières ainsi que leurs dérivées, Ann. Sci. Clermont 18 (1979), 47-64. · Zbl 0427.13006 [15] Jean-Luc Chabert, Un anneau de Prüfer, J. Algebra 107 (1987), no. 1, 1 – 16 (French, with English summary). · Zbl 0635.13004 [16] J.-L. Chabert and G. Gerboud, Polynômes à valeurs entières et binômes de Fermat, Prépublication du Labo. Math. Marseille No. 90-9. 1990. [17] Robert Gilmer, Sets that determine integer-valued polynomials, J. Number Theory 33 (1989), no. 1, 95 – 100. · Zbl 0695.13015 [18] -, Multiplicative ideal theory, Dekker, New York, 1970. [19] Robert Gilmer, William Heinzer, and David Lantz, The Noetherian property in rings of integer-valued polynomials, Trans. Amer. Math. Soc. 338 (1993), no. 1, 187 – 199. · Zbl 0780.13009 [20] Donald L. McQuillan, On Prüfer domains of polynomials, J. Reine Angew. Math. 358 (1985), 162 – 178. · Zbl 0568.13003 [21] Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York-London, 1962. · Zbl 0123.03402 [22] A. Ostrowski, Über ganzwertige polynome in algebraischen Zahlkörpern, J. Reine Angew. Math. 149 (1919), 117-124. · JFM 47.0163.05 [23] G. Polya, Über ganzwertige polynome in algebraischen Zahlkörpern, J. Reine Angew. Math. 149 (1919), 97-116. · JFM 47.0163.04 [24] Gérard Rauzy, Ensembles arithmétiquement denses, C. R. Acad. Sci. Paris Sér. A-B 265 (1967), A37 – A38 (French). · Zbl 0153.07903
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