In algebra,

Hua Hua or HUA may refer to: China * Hua, as in Huaxia and Zhonghua, a name of China ** Hoa people, Chinese people in Vietnam * Hua (state), a state in ancient China, destroyed by Qin * Hua (surname), a Chinese surname * Hua County, in Anyang, Hena ...

's identity named after Hua Luogeng, states that for any elements ''a'', ''b'' in a

division ring In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...

a - \left(a^ + \left(b^ - a\right)^\right)^ = aba

whenever

ab \ne 0, 1

. Replacing

b

with

-b^

gives another equivalent form of the identity:

\left(a + ab^a\right)^ + (a + b)^ = a^.

Hua's theorem

The identity is used in a proof of Hua's theorem, which states that if

\sigma

is a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

between division rings satisfying

\sigma(a + b) = \sigma(a) + \sigma(b), \quad \sigma(1) = 1, \quad \sigma(a^) = \sigma(a)^,

then

\sigma

is a

homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...

or an

antihomomorphism In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is a bijective antihomomorphism, i.e. an antiisomorphism, from a set to itself. From ...

. This theorem is connected to the

fundamental theorem of projective geometry In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...

Proof of the identity

One has

(a - aba)\left(a^ + \left(b^ - a\right)^\right) = 1 - ab + ab\left(b^ - a\right)\left(b^ - a\right)^ = 1.

The proof is valid in any ring as long as

a, b, ab - 1

are

unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...

References

* * Theorems in algebra {{algebra-stub