On S -2-Prime Ideals of Commutative Rings

Prime ideals and their generalizations are crucial in numerous research areas, particularly in commutative algebra. The concept of generalization of prime ideals begins with the study of weakly prime ideals. Since then, subsequent works aimed at expanding this concept into more generalized forms. Among these, S -prime ideals and 2-prime ideals have reaped attention recently. This paper aims to characterize S -2-prime ideals, which serve as a generalization encompassing both 2-prime ideals and S -prime ideals. To accomplish this objective, we construct an ideal which distinct from a multiplicatively closed subset with the help of commutative rings. We investigate the localization and the S -2-prime avoidance lemma in commutative rings. Furthermore, we explore the properties of this class of ideals in trivial ring extensions and amalgamated algebras along an ideal. We delve into S -properties for compactly packedness, compactly 2-packedness and coprimely packedness in trivial ring extentions. Moreover, this notion of ideals helps us to indicate that many results stated in S -prime ideals and 2-prime ideals can be readily expanded to the framework of S -2-prime ideals. Supporting examples also highlight a significant distinction between S -2-prime ideals and stated ideals.