# 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 |