In algebra, Hua's identity[1] named after Hua Luogeng, states that for any elements a, b in a division ring,
+
Replacing
gives another equivalent form of the identity:
The identity is used in a proof of Hua's theorem,[2] which states that if
σ
is a function between division rings satisfying
σ ( a + b ) = σ ( a ) + σ ( b ) ,
σ ( 1 ) = 1 ,
σ (
) = σ ( a
σ
is a homomorphism or an antihomomorphism.
This theorem is connected to the fundamental theorem of projective geometry.
{\displaystyle (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
are units.
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