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

Abstract 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 $$S\subseteq R$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> . 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 $$A\times _{C}B$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:msub> <mml:mo>×</mml:mo> <mml:mi>C</mml:mi> </mml:msub> <mml:mi>B</mml:mi> </mml:mrow> </mml:math> is obtained, showing that the property transfers precisely through the coordinate ideals together with a natural square-difference compatibility condition.