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

TL;DR

This paper characterizes Jordan mappings on various rings and algebras through zero product conditions, showing such maps decompose into a Jordan derivation and a multiplier.

Contribution

It establishes a general form for additive maps satisfying zero product conditions on generalized matrix rings and applies this to several classes of algebras.

Findings

Additive maps decompose into Jordan derivations and multipliers.

Results apply to matrix algebras, prime rings, and operator algebras.

Zero product conditions characterize Jordan mappings.

Abstract

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.