Weakly S-square-difference factor absorbing ideals in commutative rings

We develop the theory of weakly S -square-difference factor absorbing ideals (weakly S -sdf ideals) in a commutative ring R , extending the notion of weakly sdf-absorbing ideals to a relative setting determined by a multiplicatively closed subset S⊆ R S ⊆ R . We establish their basic properties, relate them to weakly S -prime, S -sdf-absorbing, and S -semiprime ideals, and use the S -characteristic and S -invertible elements to identify conditions under which weakly S -sdf ideals strengthen to weakly S -prime or S -sdf-absorbing ideals. We further examine their behavior under homomorphic images, direct products, quotients, trivial ring extensions, amalgamated duplications, and amalgamated algebras. A complete characterization of weakly S -sdf ideals in pullback rings A× CB A × C B is obtained, showing that the property transfers precisely through the coordinate ideals together with a natural square-difference compatibility condition.