Hua's identity

In algebra, Hua's identity[1] states that for any elements a, b in a division ring,

${\displaystyle a-(a^{-1}+(b^{-1}-a)^{-1})^{-1}=aba}$

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

whenever ${\displaystyle ab\neq 0,1}$. Replacing ${\displaystyle b}$ with ${\displaystyle -b^{-1}}$ gives another equivalent form of the identity:

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

An important application of the identity is a proof of Hua's theorem.[2][3] The theorem says that if ${\displaystyle \sigma :K\to L}$ is a function between division rings and if ${\displaystyle \sigma }$ satisfies:

${\displaystyle \sigma (a+b)=\sigma (a)+\sigma (b),\quad \sigma (1)=1,\quad \sigma (a^{-1})=\sigma (a)^{-1},}$

then ${\displaystyle \sigma }$ is either a homomorphism or an antihomomorphism. The theorem is important because of the connection to the fundamental theorem of projective geometry.

Proof

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

(Note the proof is valid in any ring as long as ${\displaystyle a,b,ab-1}$ are units.[4])

References

1. Cohn 2003, §9.1
2. Cohn 2003, Theorem 9.1.3
3. "Is this map of domains a Jordan homomorphism?". math.stackexchange.com. Retrieved 2016-06-28.
4. Jacobson, § 2.2. Exercise 9. harvnb error: no target: CITEREFJacobson (help)

• 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.
• Jacobson, Nathan (2009), Basic Algebra 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1