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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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