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

Abstract

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

16 pages