Finite generation properties for various rings of integer-valued polynomials. (English) Zbl 1177.13051

If \(D\) is an integral domain with quotient field \(K\), \(D\neq K\) and \(E\subset K\), then \(Int(E,D)\) is the set of all \(f\in K[X]\) with \(f(E)\subset D\). The authors study two classes of subrings of \(Int(E,D)\): \(Int^{(r)}(E,D)\) consisting of all polynomials which lie in \(Int(E,D)\) with all their divided differences of order up to \(r\), and \(Int_x(E,D)=\{f\in K[X]:\;f(xX+a)\in D[X]\;\text{for\;all}\;a\in E\}\) [introduced earlier by the third author in her Ph.D. thesis, cf. Commun. Algebra 32, No. 8, 3043–3069 (2004; Zbl 1061.13011)]. After establishing the main properties of these rings the authors show that if \(D\) is a Dedekind domain and \(x\neq0\), then the ring \(Int_x(E,D)\) is noetherian, and if \(D\) is a Dedekind domain with finite residue fields, then the rings \(Int^{(r)}(E,D)\) are not noetherian. It is also shown that if \(D\) is either a Krull domain or a noetherian integrally closed domain, then for a class of subsets \(E\) of \(D\) the ring \(Int_x(E,D)\) is noetherian.


13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13B25 Polynomials over commutative rings
13E05 Commutative Noetherian rings and modules
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13F30 Valuation rings


Zbl 1061.13011
Full Text: DOI


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