Let R be a commutative ring with a nonzero identity. We introduce (weakly) cube-difference factor absorbing semiprimary ideals, which are proper ideals I of R satisfying, for all elements a and b of R with <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(0 \ne ) \, a^{3} - b^{3} \in I$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>≠</mo> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> <msup> <mi>a</mi> <mn>3</mn> </msup> <mo>-</mo> <msup> <mi>b</mi> <mn>3</mn> </msup> <mo>∈</mo> <mi>I</mi> </mrow> </math></EquationSource> </InlineEquation>, either <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$a^{2} +ab + b^{2} \in \sqrt{I}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>a</mi> <mi>b</mi> <mo>+</mo> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>∈</mo> <msqrt> <mi>I</mi> </msqrt> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$a - b \in \sqrt{I}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>-</mo> <mi>b</mi> <mo>∈</mo> <msqrt> <mi>I</mi> </msqrt> </mrow> </math></EquationSource> </InlineEquation>. Among various results and structural properties, several supporting examples are given. Further, we investigate the submodule’s counterpart of the cdf-absorbing ideal concept and examine their relationship across the idealization construction of modules. Moreover, we examine the structure of the introduced class of ideals across several ring constructions, including polynomial rings, localization rings, amalgamated rings, and product rings.

On cube-difference factor absorbing semiprimary ideals

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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 a nonzero identity. We introduce (weakly) cube-difference factor absorbing semiprimary ideals, which are proper ideals I of R satisfying, for all elements a and b of R with $(0 \ne ) \, a^{3} - b^{3} \in I$ $(0 \neq) a^{3} - b^{3} \in I$ , either $a^{2} +ab + b^{2} \in \sqrt{I}$ $a^{2} + a b + b^{2} \in \sqrt{I}$ or $a - b \in \sqrt{I}$ $a - b \in \sqrt{I}$ . Among various results and structural properties, several supporting examples are given. Further, we investigate the submodule’s counterpart of the cdf-absorbing ideal concept and examine their relationship across the idealization construction of modules. Moreover, we examine the structure of the introduced class of ideals across several ring constructions, including polynomial rings, localization rings, amalgamated rings, and product rings.