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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 similarconclusion remains valid on full matrix algebras, unital prime rings with anontrivial idempotent, unital standard operator algebras, CDCSL algebras andvon Neumann algebras.

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