Hua's identity

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

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

An important application of the identity is a proof of Hua's theorem. The theorem says that if ${\displaystyle \sigma :K\to L}$ is a function between division rings and if ${\displaystyle \sigma }$ satisfies:

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