Ore domain

Let $R$ be a domain (http://planetmath.org/IntegralDomain). We say that $R$ is a right Ore domain if any two nonzero elements of $R$ have a nonzero common right multiple, i.e. for every pair of nonzero $x$ and $y$, there exists a pair of elements $r$ and $s$ of $R$ such that $xr=ys\ne 0$.

This condition turns out to be equivalent to the following conditions on $R$ when viewed as a right $R$-module:
(a) $R_R$ is a uniform module.
(b) $R_R$ is a module of finite rank.

The definition of a left Ore domain is similar.

If $R$ is a commutative domain (http://planetmath.org/IntegralDomain), then it is a right (and left) Ore domain.

Title	Ore domain
Canonical name	OreDomain
Date of creation	2013-03-22 11:51:17
Last modified on	2013-03-22 11:51:17
Owner	antizeus (11)
Last modified by	antizeus (11)
Numerical id	11
Author	antizeus (11)
Entry type	Definition
Classification	msc 16S10