Let D be an integral domain with quotient field K and E a subset of K. The ring of integer-valued rational functions on E is defined as <Equation ID="Equ1"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13_2024_2086_Article_Equ1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="288" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \mathrm {Int^R}(E,D):=\lbrace \varphi \in K(X);\; \varphi (E)\subseteq D\rbrace . \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msup> <mi mathvariant="normal">Int</mi> <mi mathvariant="normal">R</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mi>φ</mi> <mo>∈</mo> <mi>K</mi> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>;</mo> <mspace width="0.277778em" /> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> <mo>⊆</mo> <mi>D</mi> <mo stretchy="false">}</mo> </mrow> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>The main goal of this paper is to investigate the Krull dimension of the ring <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2024_2086_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="84" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathrm {Int^R}(E,D).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="normal">Int</mi> <mi mathvariant="normal">R</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Particularly, we are interested in domains that are either Jaffard or PVDs. Interesting results are established with some illustrating examples.

On the Krull dimension of rings of integer-valued rational functions

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发表日期：

2025年1月3日

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M. M. Chems-Eddin,
B. Feryouch,
H. Mouanis,
A. Tamoussit

摘要

Let D be an integral domain with quotient field K and E a subset of K. The ring of integer-valued rational functions on E is defined as $\begin{aligned} \mathrm {Int^R}(E,D):=\lbrace \varphi \in K(X);\; \varphi (E)\subseteq D\rbrace . \end{aligned}$ $\begin{matrix} {Int}^{R} (E, D) : = {φ \in K (X); φ (E) \subseteq D} . \end{matrix}$ The main goal of this paper is to investigate the Krull dimension of the ring $\mathrm {Int^R}(E,D).$ ${Int}^{R} (E, D) .$ Particularly, we are interested in domains that are either Jaffard or PVDs. Interesting results are established with some illustrating examples.