Hua's identity

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

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

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

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

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

Hua's theorem

The identity is used in a proof of Hua's theorem,^[2] which states that if ${\displaystyle \sigma }$  is a function between division rings satisfying

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

 

then ${\displaystyle \sigma }$  is a homomorphism or an antihomomorphism. This theorem is connected to the fundamental theorem of projective geometry.

Proof of the identity

One has

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

 

The proof is valid in any ring as long as ${\displaystyle a,b,ab-1}$  are units.^[3]

References

1. Cohn 2003, §9.1
2. Cohn 2003, Theorem 9.1.3
3. Jacobson 2009, § 2.2. Exercise 9.

• 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. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-47189-1. OCLC 294885194.