Hua's identity

Short description: Formula relating pairs of elements in a division ring

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

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

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. 
• Jacobson, Nathan (2009). Basic algebra. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-47189-1. OCLC 294885194.