Work

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

Year: 2026

Type: article

Abstract: 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.

Source: Ricerche di Matematica

DOI: 10.1007/s11587-026-01085-w

Authors: Alaa Abouhalaka, Nico Groenewald

Institutions: Ministry of Education and Higher Education, Cukurova University, Nelson Mandela University

Language: English

Topic: Rings, Modules, and Algebras

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,190

Funder: Qatar National Library