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

Abstract This paper introduces and investigates new generalizations of both 1-absorbing prime and $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> -ideals in the setting of noncommutative rings with identity, using special radicals $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> . We define the notions of $$(1,\rho )$$ <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 $$(1,\rho ^*)$$ <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 for a given special radical $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> , 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 $$\rho = \mathcal {P}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ρ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>P</mml:mi> </mml:mrow> </mml:math> , the prime radical, where we establish specific properties in various classes of rings, including CI -rings.