Hua's identity

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

In algebra, Hua's identity named after Hua Luogeng, states that for any elements a, b in a division ring, $$a-\left(a^{-1}+\left(b^{-1}-a\right)^{-1}\right)^{-1}=aba$$ whenever $ab\neq 0,1$. Replacing $b$ with $-b^{-1}$ gives another equivalent form of the identity: $$\left(a+ab^{-1}a\right)^{-1}+(a+b)^{-1}=a^{-1}.$$

Hua's theorem

The identity is used in a proof of Hua's theorem, which states that if $\sigma$ is a function between division rings satisfying $$\sigma (a+b)=\sigma (a)+\sigma (b),\quad \sigma (1)=1,\quad \sigma (a^{-1})=\sigma (a)^{-1},$$ then $\sigma$ is a homomorphism or an antihomomorphism. This theorem is connected to the fundamental theorem of projective geometry.

Proof of the identity

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

The proof is valid in any ring as long as $a,b,ab-1$ 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.
• Jacobson, Nathan (2009). Basic algebra. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-47189-1. OCLC 294885194.