Hua's identity

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

$a-(a^{{-1}}+(b^{{-1}}-a)^{{-1}})^{{-1}}=aba$

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

$(a+ab^{{-1}}a)^{{-1}}+(a+b)^{{-1}}=a^{{-1}}.$

An important application of the identity is a proof of Hua's theorem.^[2][3] The theorem says that if $\sigma :K\to L$ is a function between division rings and if $\sigma$ satisfies:

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

then $\sigma$ is either a homomorphism or an antihomomorphism. The theorem is important because of the connection to the fundamental theorem of projective geometry.

Proof

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

References

1. Cohn 2003, §9.1
2. Cohn 2003, Theorem 9.1.3
3. "Is this map of domains a Jordan homomorphism?". math.stackexchange.com. Retrieved 2016-06-28.

• 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.