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<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1192_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="104" /> </InlineMediaObject> <EquationSource Format="TEX">\(R=\bigoplus _{\alpha \in \Gamma } R_{\alpha }\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo>=</mo> <msub> <mo>⨁</mo> <mrow> <mi>α</mi> <mo>∈</mo> <mi mathvariant="normal">Γ</mi> </mrow> </msub> <msub> <mi>R</mi> <mi>α</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> be a commutative ring graded by any arbitrary torsionless grading monoid <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1192_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>. In this article, we introduce the graded version of the notion of Prüfer rings, which is a generalization of graded Prüfer domains to the context of arbitrary <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1192_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-graded rings with zero-divisors and we extend some properties to the graded situation.</p>

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Graded Prüfer rings

  • 期刊论文

  • 发表日期：

    2025年2月18日

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  • Nassima Guennach,
  • Najib Mahdou,
  • Abdelkbir Riffi

摘要

Let \(R=\bigoplus _{\alpha \in \Gamma } R_{\alpha }\) R = ⨁ α ∈ Γ R α be a commutative ring graded by any arbitrary torsionless grading monoid \(\Gamma \) Γ . In this article, we introduce the graded version of the notion of Prüfer rings, which is a generalization of graded Prüfer domains to the context of arbitrary \(\Gamma \) Γ -graded rings with zero-divisors and we extend some properties to the graded situation.