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	2013-03-22 11:48:46
Last modified on	2013-03-22 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