Let R be a commutative ring with nonzero identity, and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation> an expansion function of its ideals. As a generalization of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>-n-ideals, we introduce weakly <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>-n-ideals which are proper ideals I of R for which, whenever <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$0 \ne xy \in I$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≠</mo> <mi>x</mi> <mi>y</mi> <mo>∈</mo> <mi>I</mi> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$x, y \in R$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>R</mi> </mrow> </math></EquationSource> </InlineEquation>, it follows that <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$x \in \delta (I)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> or y is nilpotent. By exploring various basic properties of weakly <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>-n-ideals and establishing several characterizations, we reveal the principal symmetries they share with <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>-n-ideals and related classes such as n-ideals, <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\delta (0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-ideals, and <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>-semiprimary ideals. The transfer of the newly introcued property to ideals of quotient, localization, and product rings is also examined.

Weakly $\delta $-n-ideals of commutative rings

期刊论文
发表日期：

2026年4月7日

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Khalid Draoui

摘要

Let R be a commutative ring with nonzero identity, and $\delta $ $δ$ an expansion function of its ideals. As a generalization of $\delta $ $δ$ -n-ideals, we introduce weakly $\delta $ $δ$ -n-ideals which are proper ideals I of R for which, whenever $0 \ne xy \in I$ $0 \neq x y \in I$ for $x, y \in R$ $x, y \in R$ , it follows that $x \in \delta (I)$ $x \in δ (I)$ or y is nilpotent. By exploring various basic properties of weakly $\delta $ $δ$ -n-ideals and establishing several characterizations, we reveal the principal symmetries they share with $\delta $ $δ$ -n-ideals and related classes such as n-ideals, $\delta (0)$ $δ (0)$ -ideals, and $\delta $ $δ$ -semiprimary ideals. The transfer of the newly introcued property to ideals of quotient, localization, and product rings is also examined.

Weakly \(\delta \)-n-ideals of commutative rings

摘要