Let R be a commutative ring with nonzero unit, and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation> an expansion function of its ideals. In this paper, we introduce sdf-absorbing <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>-primary ideals. A proper ideal I of R is termed square-difference factor absorbing <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>-primary (sdf-absorbing <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>-primary) if, for all <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$0 \ne a, b \in R$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≠</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>R</mi> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq8"> <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>, it follows that <InlineEquation ID="IEq9"> <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="IEq10"> <EquationSource Format="TEX">$a - b \in \delta (I)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>-</mo> <mi>b</mi> <mo>∈</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Several properties and results are presented and supported by illustrative examples showing, in particular, the nontrivial nature of the introduced class. Moreover, we examine the transfer of sdf-absorbing <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>-primary ideals under ring homomorphisms, and their behavior across various fundamental ring-theoretic constructions, including localization rings, polynomial rings, product rings, trivial ring extensions and amalgamated rings.

On square-difference factor absorbing $\delta $-primary ideals of commutative rings

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发表日期：

2025年12月2日

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Khalid Draoui

摘要

Let R be a commutative ring with nonzero unit, and $\delta $ $δ$ an expansion function of its ideals. In this paper, we introduce sdf-absorbing $\delta $ $δ$ -primary ideals. A proper ideal I of R is termed square-difference factor absorbing $\delta $ $δ$ -primary (sdf-absorbing $\delta $ $δ$ -primary) if, for all $0 \ne a, b \in R$ $0 \neq a, b \in R$ with $a^2 - b^2 \in I$ $a^{2} - b^{2} \in I$ , it follows that $a + b \in I$ $a + b \in I$ or $a - b \in \delta (I)$ $a - b \in δ (I)$ . Several properties and results are presented and supported by illustrative examples showing, in particular, the nontrivial nature of the introduced class. Moreover, we examine the transfer of sdf-absorbing $\delta $ $δ$ -primary ideals under ring homomorphisms, and their behavior across various fundamental ring-theoretic constructions, including localization rings, polynomial rings, product rings, trivial ring extensions and amalgamated rings.

On square-difference factor absorbing \(\delta \)-primary ideals of commutative rings

摘要