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In algebra, Hua's identity^[1] named after Hua Luogeng, states that for any elements a, b in a division ring, $a-\left(a^{-1}+\left(b^{-1}-a\right)^{-1}\right)^{-1}=aba$ whenever $ab\neq 0,1$ . Replacing $b$ with $-b^{-1}$ gives another equivalent form of the identity: $\left(a+ab^{-1}a\right)^{-1}+(a+b)^{-1}=a^{-1}.$

Hua's theorem

The identity is used in a proof of Hua's theorem,^[2] which states that if $\sigma$ is a function between division rings satisfying $\sigma (a+b)=\sigma (a)+\sigma (b),\quad \sigma (1)=1,\quad \sigma (a^{-1})=\sigma (a)^{-1},$ then $\sigma$ is a homomorphism or an antihomomorphism. This theorem is connected to the fundamental theorem of projective geometry.

Proof of the identity

One has $(a-aba)\left(a^{-1}+\left(b^{-1}-a\right)^{-1}\right)=1-ab+ab\left(b^{-1}-a\right)\left(b^{-1}-a\right)^{-1}=1.$

The proof is valid in any ring as long as $a,b,ab-1$ are units.^[3]

References

^ Cohn 2003, §9.1
^ Cohn 2003, Theorem 9.1.3
^ Jacobson 2009, § 2.2. Exercise 9.

Cohn, Paul M. (2003). Further algebra and applications (Revised ed. of Algebra, 2nd ed.). London: Springer-Verlag. ISBN 1-85233-667-6. Zbl 1006.00001.
Jacobson, Nathan (2009). Basic algebra. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-47189-1. OCLC 294885194.

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[1] Cohn 2003, §9.1

[2] Cohn 2003, Theorem 9.1.3

[3] Jacobson 2009, § 2.2. Exercise 9.

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[2]

[3]

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+{{short description|Formula relating pairs of elements in a division ring}}
 {{for|Hua's identity in [[Jordan algebra]]s|Hua's identity (Jordan algebra)}}
-In algebra, '''Hua's identity'''<ref>{{harvnb|Cohn|2003|loc=§9.1}}</ref> <!-- named after --> states that for any elements ''a'', ''b'' in a [[division ring]],
+In algebra, '''[[Hua Luogeng|Hua]]'s identity'''<ref>{{harvnb|Cohn|2003|loc=§9.1}}</ref> named after Hua Luogeng, states that for any elements ''a'', ''b'' in a [[division ring]],
-:<math>a - (a^{-1} + (b^{-1} - a)^{-1})^{-1} = aba</math>
+<math display="block">a - \left(a^{-1} + \left(b^{-1} - a\right)^{-1}\right)^{-1} = aba</math>
 whenever <math>ab \ne 0, 1</math>. Replacing <math>b</math> with <math>-b^{-1}</math> gives another equivalent form of the identity:
-:<math>(a+ab^{-1}a)^{-1} + (a+b)^{-1} =a^{-1}.</math>
+<math display="block">\left(a + ab^{-1}a\right)^{-1} + (a + b)^{-1} = a^{-1}.</math>
+==Hua's theorem==
-An important application of the identity is a proof of '''Hua's theorem'''.<ref>{{harvnb|Cohn|2003|loc=Theorem 9.1.3}}</ref><ref>http://math.stackexchange.com/questions/161301/is-this-map-of-domains-a-jordan-homomorphism</ref> The theorem says that if <math>\sigma: K \to L</math> is a [[function (mathematics)|function]] between division rings and if <math>\sigma</math> satisfies:
+The identity is used in a proof of '''Hua's theorem''',<ref>{{harvnb|Cohn|2003|loc=Theorem 9.1.3}}</ref> which states that if <math>\sigma</math> is a [[function (mathematics)|function]] between division rings satisfying
-:<math>\sigma(a + b) = \sigma(a) + \sigma(b), \quad \sigma(1) = 1, \quad \sigma(a^{-1}) = \sigma(a)^{-1},</math>
+<math display="block">\sigma(a + b) = \sigma(a) + \sigma(b), \quad \sigma(1) = 1, \quad \sigma(a^{-1}) = \sigma(a)^{-1},</math>
-then <math>\sigma</math> is either a [[Ring homomorphism|homomorphism]] or an antihomomorphism. The theorem is important because of the connection to the [[fundamental theorem of projective geometry]].
+then <math>\sigma</math> is a [[Ring homomorphism|homomorphism]] or an [[antihomomorphism]]. This theorem is connected to the [[fundamental theorem of projective geometry]].
-== Proof ==
+== Proof of the identity==
+One has
+<math display="block">(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.</math>
+The proof is valid in any ring as long as <math>a, b, ab - 1</math> are [[unit (ring theory)|unit]]s.<ref>{{harvnb|Jacobson|2009|loc=§ 2.2. Exercise 9.}}</ref>
-: <math>(a - aba)(a^{-1} + (b^{-1} - a)^{-1}) = ab(b^{-1} - a)(a^{-1} +  (b^{-1} - a)^{-1}) = 1.</math>
 == References ==
 {{reflist}}
+{{refbegin}}
 * {{cite book | first=Paul M. | last=Cohn | authorlink=Paul Cohn | edition=Revised ed. of Algebra, 2nd | title=Further algebra and applications | year=2003 | location=London | publisher=[[Springer-Verlag]] | isbn=1-85233-667-6 | zbl=1006.00001 }}
+* {{cite book | last=Jacobson | first=Nathan | title=Basic algebra | publisher=Dover Publications | publication-place=Mineola, N.Y. | date=2009 | isbn=978-0-486-47189-1 | oclc=294885194 }}
+{{refend}}
 [[Category:Theorems in algebra]]
+[[Category:Hua Luogeng]]
 {{algebra-stub}}