Hua's identity

For Hua's identity in Jordan algebras, see Hua's identity (Jordan algebra).

In algebra, Hua's identity^[1] named after Hua Luogeng, states that for any elements a, b in a division ring, $${\displaystyle a-\left(a^{-1}+\left(b^{-1}-a\right)^{-1}\right)^{-1}=aba}$$ whenever ${\displaystyle ab\neq 0,1}$. Replacing ${\displaystyle b}$ with ${\displaystyle -b^{-1}}$ gives another equivalent form of the identity: $${\displaystyle \left(a+ab^{-1}a\right)^{-1}+(a+b)^{-1}=a^{-1}.}$$

Hua's theorem

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.

Proof of the identity

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]