# Goldie ring

Let $R$ be a ring. If the set of annihilators^{}
$\{\mathrm{r}.\mathrm{ann}(x)\mid x\in R\}$
satisifies the ascending chain condition^{}, then
$R$ is said to satisfy the *ascending chain condition on right annihilators*.

A ring $R$ is called a *right Goldie ring* if it satisfies the ascending chain condition on right annihilators and ${R}_{R}$ is a module of finite rank.

*Left Goldie ring* is defined similarly. If the context makes it clear on which side the ring operates, then such a ring is simply called a *Goldie ring*.

A right Noetherian ring is right Goldie.

Title | Goldie ring |

Canonical name | GoldieRing |

Date of creation | 2013-03-22 14:04:10 |

Last modified on | 2013-03-22 14:04:10 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 6 |

Author | mclase (549) |

Entry type | Definition |

Classification | msc 16P60 |

Synonym | Goldie |

Related topic | UniformDimension |

Defines | left Goldie |

Defines | right Goldie |

Defines | left Goldie ring |

Defines | right Goldie ring |