Hua's identity

In algebra, Hua's identity states that for any elements a, b in a division ring,

$a-(a^{-1}+(b^{-1}-a)^{-1})^{-1}=aba$

whenever $ab\neq 0,1$.

An important application of the identity is a proof of Hua's theorem. The theorem says that if $\sigma :K\to L$ is a function between division rings and if $\sigma$ satisfies:

$\sigma (a+b)=\sigma (a)+\sigma (b),\quad \sigma (1)=1,\quad \sigma (a^{-1})=\sigma (a)^{-1},$

then $\sigma$ is either a homomorphism or an antihomomorphism.

Proof

$(a-aba)(a^{-1}+(b^{-1}-a)^{-1})=ab(b^{-1}-a)(a^{-1}+(b^{-1}-a)^{-1})=1.$

References

• Cohn, Paul M. (2003). Further algebra and applications. Springer. ISBN 1-85233-667-6.