# 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 $xr=ys\neq 0$.

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 OreDomain 2013-03-22 11:51:17 2013-03-22 11:51:17 antizeus (11) antizeus (11) 11 antizeus (11) Definition msc 16S10