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Skew field

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(Redirected from Division ring)

A skew field, also known as a division ring, is a (not necessarily commutative) ring in which every element has a two-sided inverse. Equivalently, a skew field is a field in which multiplication does not necessarily commute. That is, it is a set S along with two operations, + and \cdot such that:

  • There are elements 1, 0 \in S such that 1 \cdot a = a \cdot 1 = a and a + 0 = 0 + a = a for all a \in S. (Existence of additive and multiplicative identities.)
  • For each a \in S other than 0, there exist elements a^{-1}, -a \in S such that a\cdot a^{-1} = a^{-1}\cdot a = 1 and a + (-a) = (-a) + a = 0. (Existence of additive and multiplicative inverses.)
  • a + b = b + a for all a, b \in S (Commutativity of addition.)
  • (a + b) + c = a + (b + c) for all a, b, c \in S (Associativity of addition.)
  • (a \cdot b )\cdot c = a \cdot (b \cdot c) (Associativity of multiplication.)
  • a(b + c) = ab + ac and (b + c)a = ba + ca (The distributive property.)


Every field is a skew field. The most famous example of a skew field that is not also a field is the collection of quaternions.


See Also

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