Hua's identity

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

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

Hua's theorem

The identity is used in a proof of Hua's theorem,^[2][3] which states that if ${\displaystyle \sigma }$ is a function between division rings satisfying

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

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

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.