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<p>We study weakly (1, <i>n</i>)-ideals and weakly <i>n</i>-ideals in commutative rings. Let <i>A</i> be a commutative ring with a nonzero identity and I be a proper ideal of <i>A</i>. Then <i>I</i> is said to be a weakly (1, <i>n</i>)-ideal (or weakly <i>n</i>-ideal) if whenever 0 ≠ <i>abc</i> ∈ <i>I</i> for some nonunits <i>a,b,c</i> ∈ <i>A</i> (or 0 ≠ <i>ab</i> ∈ <i>I</i> for some <i>a,b</i> ∈ <i>A</i>), then either <i>ab</i> ∈ <i>I</i> or <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10587_2025_3724_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">\(c \in \mathfrak{N}(A)\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>c</mi> <mo>∈</mo> <mrow> <mi mathvariant="fraktur">N</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> (or <i>a</i> ∈ <i>I</i> or <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10587_2025_3724_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">\(b \in \mathfrak{N}(A)\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>b</mi> <mo>∈</mo> <mrow> <mi mathvariant="fraktur">N</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>, respectively), where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10587_2025_3724_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathfrak{N}(A)\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="fraktur">N</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> is the set of all nilpotent elements of <i>A</i>. Many examples and properties of weakly (1, <i>n</i>)-ideals and weakly <i>n</i>-ideals are given. We characterize all rings in which every proper ideal is a weakly (1, <i>n</i>)-ideal and weakly <i>n</i>-ideal. Furthermore, we investigate both weakly (1, <i>n</i>)-ideals and weakly <i>n</i>-ideals in amalgamated algebras along an ideal.</p>

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On weakly (1,n)-ideals and weakly n-ideals

  • 期刊论文

  • 发表日期：

    2025年3月28日

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  • Bayram Ali Ersoy,
  • Suat Koç,
  • Ünsal Tekır,
  • Gürsel Yeşilot,
  • Eda Yıldız

摘要

We study weakly (1, n)-ideals and weakly n-ideals in commutative rings. Let A be a commutative ring with a nonzero identity and I be a proper ideal of A. Then I is said to be a weakly (1, n)-ideal (or weakly n-ideal) if whenever 0 ≠ abc ∈ I for some nonunits a,b,c ∈ A (or 0 ≠ ab ∈ I for some a,b ∈ A), then either ab ∈ I or \(c \in \mathfrak{N}(A)\) c ∈ N ( A ) (or a ∈ I or \(b \in \mathfrak{N}(A)\) b ∈ N ( A ) , respectively), where \(\mathfrak{N}(A)\) N ( A ) is the set of all nilpotent elements of A. Many examples and properties of weakly (1, n)-ideals and weakly n-ideals are given. We characterize all rings in which every proper ideal is a weakly (1, n)-ideal and weakly n-ideal. Furthermore, we investigate both weakly (1, n)-ideals and weakly n-ideals in amalgamated algebras along an ideal.