AARP15

From Exampleproblems

Jump to: navigation, search

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 \isin Z, a^{-1}\isin Z\,.

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



Main Page : Abstract Algebra : Rings

Retrieved from "http://www.exampleproblems.com/wiki/index.php/AARP15"
Argan Oil
Natural Skin Care
Organic Skin Care
visitor stats