I ran into Hua's identity without intending to, meaning that I do not have a concrete reference available, and my background is not in Ring Theory.


It is apparent to me that the identity is something of a big deal, but I couldn't find any explanation why. Authors pretty much assume that if you are reading about it, you already have a feeling for how important it is. Can you give me a hint?



Motivation for Hua's identity
Hua's identity is used to prove that any additive map of a division ring into itself preserving inverses must be an automorphism or antiautomorphism. His identity puts the Jordan triple product $aba$ in terms of additions and inverses, hence showing that those maps are also Jordan automorphisms; but Jordan isomorphisms between simple algebras had been shown by Ancochea and Kaplansky (1947) to be either automorphisms or antiautomorphisms, and the result for general division rings was proved by Hua himself (1949), completing the proof for additive maps preserving inverses. The study of contexts in which Jordan homomorphisms $f:R\rightarrow S$ for $R,S$ associative rings can be described in terms of (associative) homomorphisms and antihomomorphisms has a long and rich history (e.g., Jacobson and Rickart for $R$ a matrix ring, Bresar for $S$ a semiprime ring).


1. *Le théorème de von Staudt en géométrie projective quaternionienne* (1942). Ancochea.
2. *On semi-automorphisms of division algebras* (1947). Ancochea.
3. *Semi-automorphisms of rings* (1947). Kaplansky.
4. *On the automorphisms of a sﬁeld* (1949). Hua.
5. *Jordan homomorphisms of rings* (1950). Jacobson, Rickart.
6. *Jordan mappings of semiprime rings I, II* (1989-91). Bresar.


