# Ore condition

A ring $R$ satisfies the *left Ore condition* (resp. *right Ore condition*) if and only if for all elements $x$ and $y$ with $x$ regular^{}, there exist elements $u$ and $v$ with $v$ regular such that

$$ux=vy\mathit{\hspace{1em}}\text{(resp.}xu=yv\text{).}$$ |

A ring which satisfies the (left, right) Ore condition is called a (*left*, *right*) *Ore ring*.

Title | Ore condition |

Canonical name | OreCondition |

Date of creation | 2013-03-22 14:03:04 |

Last modified on | 2013-03-22 14:03:04 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 6 |

Author | mclase (549) |

Entry type | Definition |

Classification | msc 16U20 |

Related topic | ClassicalRingOfQuotients |

Related topic | OresTheorem2 |

Defines | Ore ring |

Defines | left Ore condition |

Defines | right Ore condition |

Defines | left Ore ring |

Defines | right Ore ring |