# discrete valuation ring

A *discrete valuation ring* $R$ is a principal ideal domain^{} with exactly one nonzero maximal ideal^{} $M$. Any generator $t$ of $M$ is called a *uniformizer* or *uniformizing element* of $R$; in other words, a uniformizer of $R$ is an element $t\in R$ such that $t\in M$ but $t\notin {M}^{2}$.

Given a discrete valuation ring $R$ and a uniformizer $t\in R$, every element $z\in R$ can be written uniquely in the form $u\cdot {t}^{n}$ for some unit $u\in R$ and some nonnegative integer $n\in \mathbb{Z}$. The integer $n$ is called the *order* of $z$, and its value is independent of the choice of uniformizing element $t\in R$.

Title | discrete valuation ring |

Canonical name | DiscreteValuationRing |

Date of creation | 2013-03-22 12:16:40 |

Last modified on | 2013-03-22 12:16:40 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 9 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 13F30 |

Classification | msc 13H10 |

Synonym | DVR |

Related topic | LocalRing |

Related topic | DiscreteValuation |

Related topic | Valuation^{} |

Defines | uniformizer |

Defines | uniformizing element |

Defines | order |