Characterizing Jordan Derivable Maps on Triangular Rings by Local Actions

Suppose that T = Tri (A, M, B) is a 2-torsion free triangular ring, and G = { (A, B) vertical bar AB = 0, A, B is an element of T} boolean OR { (A, X) vertical bar A is an element of T, X is an element of {P, Q}}, where P is the standard idempotent of T and Q = I - P. Let delta: T -> T be a mapping (not necessarily additive) satisfying, (A, B) is an element of G double right arrow delta(A circle B) = A degrees delta(B) + delta(A)degrees B, where A circle B = AB + BA is the Jordan product of T. We obtain various equivalent conditions for delta, specifcally, 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.