Hua's identity

In algebra, Hua's identity^[1] 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 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1