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,
![{\displaystyle a-\left(a^{-1}+\left(b^{-1}-a\right)^{-1}\right)^{-1}=aba}](https://wikimedia.org/api/rest_v1/media/math/render/svg/306abb94e21811ff3d58099a58add5a5e3551fb7)
whenever
![{\displaystyle ab\neq 0,1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b774b6b4e37e04c0d5b2f4dd1b34ea0b6ca7f7b)
. Replacing
![{\displaystyle b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
with
![{\displaystyle -b^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7821b0bda457fb7ff9df68d4bb571397561f4b3b)
gives another equivalent form of the identity:
![{\displaystyle \left(a+ab^{-1}a\right)^{-1}+(a+b)^{-1}=a^{-1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3134aa29546557d5a31ad5712f66f2e0f1c4fd1e)
Hua's theorem
The identity is used in a proof of Hua's theorem,[2] which states that if
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},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae16773a6651ae5ae98d0f7cab8c07d844e4276f)
then
![{\displaystyle \sigma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36)
is a
homomorphism or an
antihomomorphism. This theorem is connected to the
fundamental theorem of projective geometry.
Proof of the identity
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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a05417eb79fcf12cee1c3a8eb0a56abd13f0538)
The proof is valid in any ring as long as
are units.[3]
References
This page was last edited on 15 May 2024, at 05:27