Hua's identity

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

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
$\sigma(a + b) = \sigma(a) + \sigma(b), \quad \sigma(1) = 1, \quad \sigma(a^) = \sigma(a)^,$
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
$(a - aba)\left(a^ + \left(b^ - a\right)^\right) = 1 - ab + ab\left(b^ - a\right)\left(b^ - a\right)^ = 1.$

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

References

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Theorems in algebra

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