# .

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.$$

References

Cohn 2003, §9.1
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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