Formula relating pairs of elements in a division ring
In algebra, Hua's identity[1] named after Hua Luogeng, states that for any elements a, b in a division ring,
whenever
. Replacing
with
gives another equivalent form of the identity:
Hua's theorem[edit]
The identity is used in a proof of Hua's theorem,[2] which states that if is a function between division rings satisfying
then
is a
homomorphism or an
antihomomorphism. This theorem is connected to the
fundamental theorem of projective geometry.
Proof of the identity[edit]
One has
The proof is valid in any ring as long as are units.[3]
References[edit]