# PID

A *principal ideal domain ^{}* is an integral domain

^{}where every ideal is a principal ideal

^{}.

In a PID, an ideal $(p)$ is maximal if and only if $p$ is irreducible^{}
(and prime since any PID is also a UFD (http://planetmath.org/PIDsAreUFDs)).

Note that subrings of PIDs are not necessarily PIDs. (There is an example of this within the entry biquadratic field.)

Title | PID |

Canonical name | PID |

Date of creation | 2013-03-22 11:56:25 |

Last modified on | 2013-03-22 11:56:25 |

Owner | mps (409) |

Last modified by | mps (409) |

Numerical id | 13 |

Author | mps (409) |

Entry type | Definition |

Classification | msc 16D25 |

Classification | msc 13G05 |

Classification | msc 11N80 |

Classification | msc 13A15 |

Synonym | principal ideal domain |

Related topic | UFD |

Related topic | Irreducible |

Related topic | Ideal |

Related topic | IntegralDomain |

Related topic | EuclideanRing |

Related topic | EuclideanValuation |

Related topic | ProofThatAnEuclideanDomainIsAPID |

Related topic | WhyEuclideanDomains |