# 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 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).