Hua's identity
# Hua's identity

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

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

whenever $ab\neq 0,1$. Replacing $b$ with $-b^{-1}$ gives another equivalent form of the identity:

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

## Hua's theorem

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

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

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

## Proof of the identity

One has

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

The proof is valid in any ring as long as $a,b,ab-1$ are units.

