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Encyclopedic overview
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element a has a multiplicative inverse; that is, an element usually denoted a, such that a a = a a = 1. So, (right) division may be defined as a / b = a b, but this notation is avoided, as one may have a b ≠ b a.
A commutative division ring is a field. Wedderburn's little theorem asserts that all finite division rings are commutative and therefore finite fields.
Excerpted from Wikipedia’s “division ring” article, available under the CC BY-SA 4.0 licence.