division ring
A division ring is a ring $D$ with identity^{} such that

•
$1\ne 0$

•
For all nonzero $a\in D$, there exists $b\in D$ with $a\cdot b=b\cdot a=1$
Every field is a commutative^{} division ring. The Hamiltonian quaternions are an example of a division ring which is not a field.
Title  division ring 

Canonical name  DivisionRing 
Date of creation  20130322 11:48:46 
Last modified on  20130322 11:48:46 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  10 
Author  djao (24) 
Entry type  Definition 
Classification  msc 16K99 
Classification  msc 81P05 
Synonym  skew field 