Division ring

Definition

A division ring (sometimes also associative division ring) 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.