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.
