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 = a b a {\displaystyle a-\left(a^{-1}+\left(b^{-1}-a\right)^{-1}\right)^{-1}=aba}
whenever a b ≠ 0 , 1 {\displaystyle ab\neq 0,1} . Replacing b {\displaystyle b} with − b − 1 {\displaystyle -b^{-1}} gives another equivalent form of the identity:
( a + a b − 1 a ) − 1 + ( a + b ) − 1 = a − 1 . {\displaystyle \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, which states that if σ {\displaystyle \sigma } is a function between division rings satisfying
σ ( a + b ) = σ ( a ) + σ ( b ) , σ ( 1 ) = 1 , σ ( a − 1 ) = σ ( a ) − 1 , {\displaystyle \sigma (a+b)=\sigma (a)+\sigma (b),\quad \sigma (1)=1,\quad \sigma (a^{-1})=\sigma (a)^{-1},}
then σ {\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
( a − a b a ) ( a − 1 + ( b − 1 − a ) − 1 ) = 1 − a b + a b ( b − 1 − a ) ( b − 1 − a ) − 1 = 1. {\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 a , b , a b − 1 {\displaystyle a,b,ab-1} are units.
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