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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
##### Keywords:
noetherianness; integer-valued polynomials
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##### References:
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