Division ring

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Definition

A division ring (sometimes also associative division ring) or skew field is a set D, equipped with the following:

  • An (infix) binary operation +, called addition.
  • An (infix) binary operation *, called multiplication.
  • Two distinct constants 0,1 \in D, called zero and one respectively.
  • A unary operation denoted by the prefix symbol - (Called the negative or additive inverse).
  • A map {}^{-1}: D \setminus \{ 0 \} \to D \setminus \{ 0 \}

such that the following compatibility condition holds:

  • D forms an abelian group with group operation +, identity element 0, and inverse operation -.
  • * is an associative binary operation on D.
  • D \setminus \{ 0 \} forms a group with group operation *, identity element 1, and inverse operation {}^{-1}. This group is called the multiplicative group and is denoted D^\times.
  • We have left and right distributivity laws: a * (b + c) = (a * b) + (a * c) and (a + b) * c = (a * c) + (b * c) for all a,b,c \in D.

A division ring is thus like a field except that we drop the condition of commutativity of multiplication.

The term division ring is sometimes used for the more general notion of a (possibly) nonassociative division ring. Two special cases worth mentioning are power-associative division rings and alternative division rings.

Note that in some alternative conventions, the term skew field is used for the above and the term division ring is used for a skew field that is finite-dimensional as a vector space over its center (which is a field).