Let R be a commutative ring with nonzero identity, and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation> an expansion function of its ideals. In this paper, we introduce and study the concept of square-difference factor absorbing <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\delta (0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-ideals. A proper ideal I of R is called a square-difference factor absorbing <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\delta (0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-ideal (for short, an sdf-absorbing <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\delta (0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-ideal) if, for all nonzero elements a, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$b \in R$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>∈</mo> <mi>R</mi> </mrow> </math></EquationSource> </InlineEquation>, the condition <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$a^{2} - b^{2} \in I$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>∈</mo> <mi>I</mi> </mrow> </math></EquationSource> </InlineEquation> implies that <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$a + b \in I$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>∈</mo> <mi>I</mi> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$a - b \in \delta (0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>-</mo> <mi>b</mi> <mo>∈</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We establish various properties of such ideals and examine their behavior across several classical ring constructions, including localization rings, polynomial rings, trivial ring extensions, and amalgamated rings.

Sdf-absorbing $\delta (0)$-ideals of commutative rings

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发表日期：

2026年3月2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. In this paper, we introduce and study the concept of square-difference factor absorbing $\delta (0)$ $δ (0)$ -ideals. A proper ideal I of R is called a square-difference factor absorbing $\delta (0)$ $δ (0)$ -ideal (for short, an sdf-absorbing $\delta (0)$ $δ (0)$ -ideal) if, for all nonzero elements a, $b \in R$ $b \in R$ , the condition $a^{2} - b^{2} \in I$ $a^{2} - b^{2} \in I$ implies that $a + b \in I$ $a + b \in I$ or $a - b \in \delta (0)$ $a - b \in δ (0)$ . We establish various properties of such ideals and examine their behavior across several classical ring constructions, including localization rings, polynomial rings, trivial ring extensions, and amalgamated rings.

Sdf-absorbing \(\delta (0)\)-ideals of commutative rings

摘要