Hua's identity

In algebra, Hua's identity^[1] 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,^[2]^[3] 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

fundamental theorem of projective geometry

.

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.^[4]

^ Cohn 2003, §9.1
^ Cohn 2003, Theorem 9.1.3
^ "Is this map of domains a Jordan homomorphism?". math.stackexchange.com. Retrieved 2016-06-28.
^ Jacobson 2009, § 2.2. Exercise 9.

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