.
Hua's identity
In algebra, Hua's identity[1] 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.[2][3] 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
Cohn 2003, ยง9.1
Cohn 2003, Theorem 9.1.3
http://math.stackexchange.com/questions/161301/is-this-map-of-domains-a-jordan-homomorphism
Cohn, Paul M. (2003). Further algebra and applications (Revised ed. of Algebra, 2nd ed.). London: Springer-Verlag. ISBN 1-85233-667-6. Zbl 1006.00001.
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