  Let $\mathcal{U}=\left[
  \begin{array}{cc}
  \mathcal{A} & \mathcal{M}
  \mathcal{N}& \mathcal{B}
  \end{array}
  \right]$ be a generalized matrix ring, where $\mathcal{A}$ and $\mathcal{B}$
are 2-torsion free. We prove that if $\phi :\mathcal{U}\rightarrow \mathcal{U}$
is an additive mapping such that $\phi(U)\circ V+U\circ \phi(V)=0$ 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.

Operator Algebras