## 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.

### MSC:

 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
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### References:

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