# principal ideal ring

A commutative ring $R$ in which all ideals are principal (http://planetmath.org/PrincipalIdeal), i.e. (http://planetmath.org/Ie) generated by (http://planetmath.org/IdealGeneratedBy) a single ring element, is called a principal ideal ring. If $R$ is also an integral domain^{}, it is a principal ideal domain^{}.

Some well-known principal ideal rings are the ring $\mathbb{Z}$ of integers, its factor rings $\mathbb{Z}/n\mathbb{Z}$, and any polynomial ring over a field.

Title | principal ideal ring |
---|---|

Canonical name | PrincipalIdealRing |

Date of creation | 2013-03-22 14:33:16 |

Last modified on | 2013-03-22 14:33:16 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 7 |

Author | Wkbj79 (1863) |

Entry type | Definition |

Classification | msc 13F10 |

Classification | msc 13A15 |

Synonym | principal ring |

Related topic | CriterionForCyclicRingsToBePrincipalIdealRings |