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<p>Let <i>D</i> be an integral domain with quotient field <i>K</i>,&#xa0; <i>E</i> a subset of <i>K</i> and <i>X</i> an indeterminate over <i>K</i>. The set <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textrm{Int}(E,D):=\{f\in K[X];\; f(E)\subseteq D\}\)</EquationSource> </InlineEquation>, of <i>integer-valued polynomials on</i> <i>E</i> over <i>D</i>, is known to be an integral domain. The purpose of this note is to calculate the Krull dimension of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\textrm{Int}(E,D)\)</EquationSource> </InlineEquation> across various classes of integral domains <i>D</i> and specific subsets <i>E</i> of <i>D</i>. We further extend our study to the ring <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\textrm{Int}_B(E,D):=\{f\in B[X];\; f(E)\subseteq D\},\)</EquationSource> </InlineEquation> where <i>B</i> is an integral domain containing <i>D</i>.</p>

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Note on the Krull dimension of rings of integer-valued polynomials

  • 期刊论文

  • 发表日期：

    2025年10月28日

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  • M. M. Chems-Eddin,
  • B. Feryouch,
  • A. Tamoussit

摘要

Let D be an integral domain with quotient field K,  E a subset of K and X an indeterminate over K. The set \(\textrm{Int}(E,D):=\{f\in K[X];\; f(E)\subseteq D\}\) , of integer-valued polynomials on E over D, is known to be an integral domain. The purpose of this note is to calculate the Krull dimension of \(\textrm{Int}(E,D)\) across various classes of integral domains D and specific subsets E of D. We further extend our study to the ring \(\textrm{Int}_B(E,D):=\{f\in B[X];\; f(E)\subseteq D\},\) where B is an integral domain containing D.