Characterizations of Jordan mappings on some rings and algebras through zero products

Let U = A B be a generalized matrix ring, where A and B are 2-torsion free. We prove that if phi : U -> U is an additive mapping such that phi( U) o V + U o f(V) = 0 whenever UV = VU = 0, then phi = delta + eta, where delta is a Jordan derivation and eta is a multiplier. As its applications, we prove that the similar conclusion remains valid on full matrix algebras, unital prime rings with a nontrivial idempotent, unital standard operator algebras, CDCSL algebras and von Neumann algebras.