Hua's identity

The identity is used in a proof of Hua's theorem,[2] which states that if ${\displaystyle \sigma }$ is a function between division rings satisfying

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

then ${\displaystyle \sigma }$ is a homomorphism or an antihomomorphism. This theorem is connected to the fundamental theorem of projective geometry.

One has

$${\displaystyle (a-aba)\left(a^{-1}+\left(b^{-1}-a\right)^{-1}\right)=1-ab+ab\left(b^{-1}-a\right)\left(b^{-1}-a\right)^{-1}=1.}$$

The proof is valid in any ring as long as ${\displaystyle a,b,ab-1}$ are units.[3]