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An elementary proof of the two-generator property for the ring of integer-valued polynomials. (English) Zbl 1475.13034

Let \(\text{Int}(\mathbb{Z})\) be the ring of integer-valued polynomials, that is, \[\text{Int}(\mathbb{Z})=\{f(X)\in \mathbb{Q}[X]~|~f(\mathbb{Z})\subseteq \mathbb{Z}\}.\] In [R. Gilmer and W. W. Smith, J. Algebra 81, No. 1, 150–164 (1983, Zbl 0515.13016)], it is shown, by constructive methods, that \(\text{Int}(\mathbb{Z})\) has the two-generator property, that is, every finitely generated ideal of \(\text{Int}(\mathbb{Z})\) is generated by two elements. In this paper, the auothers present an elementary proof of this known result.
Reviewer: Qiao Lei (Chengdu)

MSC:

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13E15 Commutative rings and modules of finite generation or presentation; number of generators

Citations:

Zbl 0515.13016
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Full Text: DOI

References:

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