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The Noetherian property in rings of integer-valued polynomials. (English) Zbl 0780.13009
Let \(D\) denote a commutative domain with identity with field of fractions \(K\) and let \(X\) denote an indeterminate. Then the ring \(\text{Int}(D)\) of integer-valued polynomials of \(D\) is defined by \(\text{Int}(D)=\{f(X)\in K[X]:f(D)\subseteq D\}\). The authors investigate the Noetherian property for \(\text{Int}(D)\) and for its spectrum. The first theorem states that if \(D\) and \(\text{Int}(D)\) are both Noetherian and \(M'\) is a maximal ideal of the integral closure \(D'\) of \(D\), then \(M'\) has height greater than 1 or the residue field \(D'/M'\) is infinite. In particular, if \(D\) is a Noetherian domain which is one-dimensional or integrally closed then \(\text{Int}(D)\) is Noetherian if and only if \(\text{Int}(D)=D[X]\). On the other hand, if \(D\) is Noetherian and \(D'/P'\) is infinite for every height 1 prime \(P'\) in \(D'\) then \(\text{Int}(D)\subseteq D'[X]\); in particular, if, in addition \(D'\) is a finitely generated \(D\)-module, then \(\text{Int}(D)\) is Noetherian. Next, if \(D\) is a one-dimensional local domain then \(\text{Spec(Int}(D))\) is Noetherian if and only if \(D'\) has more than one maximal ideal. The question of when \(\text{Spec(Int}(D))\) is Noetherian for \(D\) not local or of higher dimension than 1 remains open. In addition, the authors give an example of a two-dimensional domain \(D\) such that \(\text{Int}(D)\) is not Noetherian but \(\text{Int}(D_ M)\) is Noetherian for every maximal ideal \(M\) of \(D\) and also give a description of the one-dimensional domains \(D\) for which \(\text{Int}(D)\subseteq\text{Int}(D')\).
It is mentioned that P.-J. Cahen has proved some related results in a forthcoming paper entitled “Polynômes à valeurs entières sur une partie”.

MSC:
13E05 Commutative Noetherian rings and modules
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13B25 Polynomials over commutative rings
13B22 Integral closure of commutative rings and ideals
13G05 Integral domains
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