P-cycle protection

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. In algebra, Hua's identity states that for any elements a, b in a division ring,

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

whenever $ab\ne 0,1$. 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 $\sigma :K\to L$ is a function between division rings and if $\sigma$ satisfies:

$\sigma (a+b)=\sigma (a)+\sigma (b),\sigma (1)=1,\sigma (a^{-1})=\sigma (a{)}^{-1},$

then $\sigma$ is either a homomorphism or an antihomomorphism. The theorem is important because of the connection to the fundamental theorem of projective geometry.

Proof

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

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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