Skew field
From AoPSWiki
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 
along with two operations, 
and 
such that:
- There are elements
such that 
and 
for all 
. (Existence of additive and multiplicative identities.)
- For each other than 0, there exist elements
such that 
and 
. (Existence of additive and multiplicative inverses.)
-
for all 
(Commutativity of addition.)
-
for all 
(Associativity of addition.)
-
(Associativity of multiplication.)
-
and 
(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.
