title: Hua's identity
text: # 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.
  
  