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

Abstract In this paper, we introduce and explore two new classes of ideals in noncommutative rings: weakly $$(1,\rho )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -absorbing and weakly $$(1,\rho ^*)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>ρ</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -absorbing ideals, where $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> is a special radical. These concepts extend the ideas of weakly 1-absorbing prime ideals and weakly $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> -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 $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> is the prime radical $$ {\mathcal {P}}(R),$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> where we establish connections between weakly $$(1,{\mathcal {P}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>P</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -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.