Hua's identity

{{#invoke:Hatnote|hatnote}} 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 $ab\neq 0,1$. Replacing $b$ with $-b^{-1}$ gives another equivalent form of the identity:

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

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. The theorem is important because of the connection to the fundamental theorem of projective geometry.

Proof

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

References

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