Weakly S -2-Absorbing Ideals in Commutative Rings

The notion of weakly Formula: see text-absorbing ideals, introduced as a natural generalization of prime ideals, has gained significant attention in commutative rings. A weakly Formula: see text-absorbing ideal of a commutative ring Formula: see text ensures that for all Formula: see text, the inclusion Formula: see text and Formula: see text implies that at least one of Formula: see text or Formula: see text belongs to Formula: see text. This paper builds upon the existing framework by introducing and exploring weakly Formula: see text-Formula: see text-absorbing ideals in the context of a multiplicatively closed subset Formula: see text of Formula: see text. An ideal Formula: see text, disjoint from Formula: see text, is defined as weakly Formula: see text-Formula: see text-absorbing if Formula: see text and Formula: see text guarantee that Formula: see text, Formula: see text, or Formula: see text for some Formula: see text. We give the fundamental properties of weakly Formula: see text-Formula: see text-absorbing ideals and investigate their behavior under various ring constructions, such as amalgamation duplications, homomorphic images, direct products, and trivial extensions. Notable results include conditions under which the ideals retain the weakly Formula: see text-Formula: see text-absorbing property and characterizations of these ideals in special commutative rings such as reduced rings and local semiprime rings which are not prime.