Characterizing Jordan Derivable Maps on Triangular Rings by Local Actions

Abstract

Suppose that T = Tri A , ℳ , ℬ is a 2-torsion free triangular ring, and S = A , B | A B = 0 , A , B ∈ T ∪ A , X | A ∈ T , X ∈ P , Q , where P is the standard idempotent of T and Q = I − P . Let δ : T ⟶ T be a mapping (not necessarily additive) satisfying, A , B ∈ S ⇒ δ A ∘ B = A ∘ δ B + δ A ∘ B , where A ∘ B = A B + B A is the Jordan product of T . We obtain various equivalent conditions for δ , specifically, we show that δ is an additive derivation. Our result generalizes various results in these directions for triangular rings. As an application, δ on nest algebras are determined.