Characterizing Jordan Derivable Maps on Triangular Rings by Local Actions

Abstract

Suppose that $\mathcal{T}=\text{Tri}\left(\mathcal{A},\mathrm{ℳ},\mathrm{ℬ}\right)$ is a 2-torsion free triangular ring, and $\mathfrak{S}=\left\{\left(A,B\right)|AB=0,A,B\in \mathcal{T}\right\}\cup \left\{\left(A,X\right)|A\in \mathcal{T},X\in \left\{P,Q\right\}\right\}$ , where $P$ is the standard idempotent of $\mathcal{T}$ and $Q=I-P$ . Let $\delta :\mathcal{T}⟶\mathcal{T}$ be a mapping (not necessarily additive) satisfying, $\left(A,B\right)\in \mathfrak{S}\Rightarrow \delta \left(A\circ B\right)=A^{\circ }\delta \left(B\right)+\delta {\left(A\right)}^{\circ }B$ , where $A\circ B=AB+BA$ is the Jordan product of $\mathcal{T}$ . We obtain various equivalent conditions for $\delta$ , specifically, we show that $\delta$ is an additive derivation. Our result generalizes various results in these directions for triangular rings. As an application, $\delta$ on nest algebras are determined.