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

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 $ϕ:\mathcal{U}\rightarrow \mathcal{U}$ is an additive mapping such that $ϕ(U)\circ V+U\circ ϕ(V)=0$ whenever $UV=VU=0,$ then $ϕ=δ+η$, where $δ$ is a Jordan derivation and $η$ 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.

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