In algebra, Hua's identity states that for any elements a, b in a division ring,
a − ( a − 1 + ( b − 1 − a ) − 1 ) − 1 = a b a whenever a b ≠ 0 , 1 . Replacing b with − b − 1 gives another equivalent form of the identity:
( a + a b − 1 a ) − 1 + ( a + b ) − 1 = a − 1 . An important application of the identity is a proof of Hua's theorem. The theorem says that if σ : K → L is a function between division rings and if σ satisfies:
σ ( a + b ) = σ ( a ) + σ ( b ) , σ ( 1 ) = 1 , σ ( a − 1 ) = σ ( a ) − 1 , then σ is either a homomorphism or an antihomomorphism. The theorem is important because of the connection to the fundamental theorem of projective geometry.
