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Hua's identity

From Wikipedia, the free encyclopedia

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

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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

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One has

The proof is valid in any ring as long as are units.[3]

References

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  1. ^ Cohn 2003, §9.1
  2. ^ Cohn 2003, Theorem 9.1.3
  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.