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 $ab\neq 0,1$. 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

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

(Note the proof is valid in any ring as long as $a,b,ab-1$ are units.)

References

• 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.