Let R be a non-commutative prime ring with Q mr (R) its maximal right ring of quotients. It is proved that the functional identity f (x, y)[x, y] = 0 for bi-additive maps f : R × R → Q mr (R) only has the trivial solution. Basing on the result, we characterize Jordan σ -derivations and generalized Jordan semiderivations in terms of derivations, σ -derivations and semiderivations.