More on the Generalized (m,n)-Jordan Derivations and Centralizers on Certain Semiprime Rings

In this paper, we give an affirmative answer to two conjectures on generalized (m, n)-Jordan derivations and generalized (m, n)-Jordan centralizers raised in Ali and Fosner (Algebra Colloq 21:411–420, 2014) and Fosner (Demonstr Math 46:254–262, 2013). Precisely, when R is a semiprime ring, we prove, under some suitable torsion restrictions, that every nonzero generalized (m, n)-Jordan derivation (resp., a generalized (m, n)-Jordan centralizer) is a derivation (resp., a two-sided centralizer).