Hua's identity

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

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 $ab \ne 0, 1$. 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

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