On Generalized Jordan Derivations and Centralizers in Torsion-Free Semiprime Rings: Structural Characterizations and Equivalences

This paper investigates the structure of generalized Jordan derivations and generalized Jordan centralizers on torsion-free semiprime rings. We establish conditions under which every nonzero generalized Jordan derivation is a derivation mapping the ring into itself. Similarly, we prove that every generalized Jordan centralizer coincides with a two-sided centralizer. The approach builds upon several supporting lemmas to demonstrate that torsion-free restrictions and semiprimeness provide sufficient conditions for such equivalences. These results extend classical findings on derivations and centralizers in associative ring theory and contribute to the structural analysis of operator mappings in algebraic systems.