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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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