Hua's identity

In algebra, Hua's identity states that for any elements a, b in a ring (with unity),

$a-(a^{-1}+(b^{-1}-a)^{-1})^{-1}=aba$

whenever $ab\neq 0,1$.

Proof

$(a-aba)(a^{-1}+(b^{-1}-a)^{-1})=ab(b^{-1}-a)(a^{-1}+(b^{-1}-a)^{-1})=1$.

References

• Cohn, Paul M. (2003). Further algebra and applications. Springer. ISBN 1-85233-667-6.