Suppose that T = Tri(A; M; B) is a 2-torsion free triangular ring, and S = (A; B) j AB = 0; A; B 2 T [ (A; X) j A 2 T ; X 2 fP; Qg ; where P is the standard idempotent of T and Q = I −P. Let δ : T ! T be a mapping (not necessarily additive) satisfying (A; B) 2 S ) δ(A ◦ B) = A ◦ δ(B) + δ(A) ◦ B; where A ◦ B = AB + BA is the Jordan product of T . We obtain various equivalent conditions for δ, specifically, we show that δ is an additive derivation. Our result generalizes various results in these directions for triangular rings. As an application, δ on nest algebras are determined
