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Hua's identity - Wikipedia

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

The identity is used in a proof of Hua's theorem,[2] which states that if {\displaystyle \sigma } is a function between division rings satisfying {\displaystyle \sigma (a+b)=\sigma (a)+\sigma (b),\quad \sigma (1)=1,\quad \sigma (a^{-1})=\sigma (a)^{-1},} then {\displaystyle \sigma } is a homomorphism or an antihomomorphism. This theorem is connected to the fundamental theorem of projective geometry.

Proof of the identity

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One has {\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 {\displaystyle a,b,ab-1} are units.[3]

  1. ^ Cohn 2003, §9.1
  2. ^ Cohn 2003, Theorem 9.1.3
  3. ^ Jacobson 2009, § 2.2. Exercise 9.