On Weakly S-(m,n)-Prime Ideals of Commutative Rings
Mathematics, vol.14, no.14, 2026 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 14 Issue: 14
- Publication Date: 2026
- Doi Number: 10.3390/math14142594
- Journal Name: Mathematics
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Aerospace Database, zbMATH, Directory of Open Access Journals, Academic Search Ultimate (EBSCO), Materials Science & Engineering Collection (ProQuest), Technology Collection (ProQuest)
- Keywords: (m,n)-prime ideals, S-(m,n)-prime ideals, weakly (m,n)-prime ideals, weakly S-(m,n)-prime ideals
- Yıldız Technical University Affiliated: Yes
Abstract
Prime ideals and their structural generalizations constitute a fundamental pillar of commutative algebra. Recently, extensions such as n-absorbing, (Formula presented.) -closed, and (Formula presented.) -prime ideals have gained significant prominence in the literature. This paper introduces and investigates the notion of weakly S- (Formula presented.) -prime ideals as a generalization of weakly (Formula presented.) -prime and S- (Formula presented.) -prime ideals, thereby providing a broader framework for studying prime ideal structures via a given multiplicatively closed subset S. A comprehensive analysis of their algebraic properties is presented, complemented by several illustrative examples to distinguish this new class from existing ones. More precisely, a characterization of weakly S- (Formula presented.) -prime ideals within the ring (Formula presented.) is established, where p denotes a prime integer. In addition, the stability and behavior of these ideals are investigated across various algebraic constructions, such as direct products, localizations, and homomorphic images. Lastly, this class of ideals is examined within the context of trivial ring extensions (idealizations), as well as amalgamated and bi-amalgamated rings. These structural characterizations enable us to demonstrate that various classical results concerning weakly (Formula presented.) -prime ideals can be naturally extended to the broader realm of weakly S- (Formula presented.) -prime ideals, while also uncovering novel properties that do not hold in the weakly (Formula presented.) -prime ideal setting.