Division ring

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 defining ingredient::abelian group with group operation $$+$$, identity element $$0$$, and inverse operation $$-$$.
 * $$*$$ is an defining ingredient::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).