<p>In this paper, we introduce and explore two new classes of ideals in noncommutative rings: weakly <InlineEquation ID="IEq7"> <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 weakly <InlineEquation ID="IEq8"> <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, where <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation> is a special radical. These concepts extend the ideas of weakly 1-absorbing prime ideals and weakly <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>-ideals. Several equivalent characterizations of these ideals are provided, along with their behavior under ring homomorphisms, idealization constructions. Special attention is given to the case when <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation> is the prime radical <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$ {\mathcal {P}}(R),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">P</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where we establish connections between weakly <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$(1,{\mathcal {P}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi mathvariant="script">P</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-absorbing ideals and other well-known ideal structures. The paper also examines these concepts in local rings and von Neumann regular rings, providing several examples to illustrate the new notions and their relationships to existing ideal theoretic concepts.</p>

On weakly $(1,\rho )$-absorbing and weakly $(1,\rho ^*)$-absorbing ideals in noncommutative rings

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2026年1月9日

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Alaa Abouhalaka,
Nico Groenewald

摘要

In this paper, we introduce and explore two new classes of ideals in noncommutative rings: weakly $(1,\rho )$ $(1, ρ)$ -absorbing and weakly $(1,\rho ^*)$ $(1, ρ^{*})$ -absorbing ideals, where $\rho $ $ρ$ is a special radical. These concepts extend the ideas of weakly 1-absorbing prime ideals and weakly $\rho $ $ρ$ -ideals. Several equivalent characterizations of these ideals are provided, along with their behavior under ring homomorphisms, idealization constructions. Special attention is given to the case when $\rho $ $ρ$ is the prime radical $ {\mathcal {P}}(R),$ $P (R),$ where we establish connections between weakly $(1,{\mathcal {P}})$ $(1, P)$ -absorbing ideals and other well-known ideal structures. The paper also examines these concepts in local rings and von Neumann regular rings, providing several examples to illustrate the new notions and their relationships to existing ideal theoretic concepts.

On weakly \((1,\rho )\)-absorbing and weakly \((1,\rho ^*)\)-absorbing ideals in noncommutative rings

摘要