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If every nonzero element of
is a unit, then is a division ring. - If it is commutative, we call it a field.
- Otherwise we call it a strictly skew field.
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Implicitly, a division ring is a Ring with Unity.
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In a division ring, it makes sense to define the multiplicative inverse. Every non-zero 2element has a multiplicative inverse
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(Fraleigh e19.23) A division ring contains exactly two idempotent elements
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A corollary of (Fraleigh 19.15) is that for
and division ring such that , then we can talk about inverses of the form . We do this as follows define
as the sum of 1’s Now we have for
And by definition
has an inverse such that The corresponding number of
’s that are needed to get to is defined as . -
(Fraleigh 24.10) Wedderburn’s Theorem: Every finite division ring is a field.
Another way to say this is that there are no finite strictly skew fields.