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.