AARP15

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Prove that the center of a division ring is a field.


A field is a commutative division ring and the center is commutative by definition. The problem is reduced to showing that the center of a division ring is also a division ring. A division ring contains multiplicative inverses for all nonzero elements, so it only has to be shown that \forall a\in Z,a^{{-1}}\in Z\,.

If a\in Z\,, then ara^{{-1}}=r\implies r=a^{{-1}}ra\implies r=(a^{{-1}})r(a^{{-1}})^{{-1}}\, so a^{{-1}}\in Z\,.



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