Jordan isomorphisms of 2-torsionfree triangular rings

We construct a class of Jordan isomorphisms from a triangular ring , and we show that if is 2-torsionfree, any Jordan isomorphism from to another ring is of this form, up to a ring isomorphism. As an application, we show that for triangular rings in a large class, any Jordan isomorphism to another ring is a direct sum of a ring isomorphism and a ring anti-isomorphism. In particular, this applies to complete upper block triangular matrix rings and indecomposable triangular rings.