<p>This paper introduces and investigates new generalizations of both 1-absorbing prime and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>-ideals in the setting of noncommutative rings with identity, using special radicals <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>. We define the notions of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$(1,\rho )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-absorbing and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$(1,\rho ^*)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <msup> <mi>ρ</mi> <mo>∗</mo> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-absorbing ideals for a given special radical <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>, and we study the relationships between these new classes of ideals and previously known ideal classes. Several structural results are obtained, characterizing when these ideals coincide or differ, especially in the existence or absence of the identity. We provide characterizations in product rings, under homomorphic images, and for constructions such as idealizations. A portion of the study is devoted to the case when <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\rho = \mathcal {P}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo>=</mo> <mi mathvariant="script">P</mi> </mrow> </math></EquationSource> </InlineEquation>, the prime radical, where we establish specific properties in various classes of rings, including <i>CI</i>-rings.</p>

On $(1,\rho )$-Absorbing and $\left( 1,\rho ^{*}\right) $-Absorbing Ideals in Noncommutative rings

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2026年4月3日

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Alaa Abouhalaka,
Nico Groenewald

摘要

This paper introduces and investigates new generalizations of both 1-absorbing prime and $\rho $ $ρ$ -ideals in the setting of noncommutative rings with identity, using special radicals $\rho $ $ρ$ . We define the notions of $(1,\rho )$ $(1, ρ)$ -absorbing and $(1,\rho ^*)$ $(1, ρ^{*})$ -absorbing ideals for a given special radical $\rho $ $ρ$ , and we study the relationships between these new classes of ideals and previously known ideal classes. Several structural results are obtained, characterizing when these ideals coincide or differ, especially in the existence or absence of the identity. We provide characterizations in product rings, under homomorphic images, and for constructions such as idealizations. A portion of the study is devoted to the case when $\rho = \mathcal {P}$ $ρ = P$ , the prime radical, where we establish specific properties in various classes of rings, including CI-rings.

On \((1,\rho )\)-Absorbing and \(\left( 1,\rho ^{*}\right) \)-Absorbing Ideals in Noncommutative rings

摘要