# valuation determined by valuation domain

###### Theorem.

Every valuation domain determines a Krull valuation of the field of fractions^{}.

Proof. Let $R$ be a valuation domain, $K$ its field of fractions and $E$ the group of units of $R$. Then $E$ is a normal subgroup^{} of the multiplicative group^{} ${K}^{*}=K\setminus \{0\}$. So we can form the factor group ${K}^{*}/E$, consisting of all cosets $aE$ where $a\in {K}^{*}$, and attach to it the additional “coset” $0E$ getting thus a multiplicative group $K/E$ equipped with zero. If $\U0001d52a=R\setminus E$ is the maximal ideal^{} of $R$ (any valuation domain has a unique maximal ideal
— cf. valuation domain is local), then we denote ${\U0001d52a}^{*}=\U0001d52a\setminus \{0\}$ and $S={\U0001d52a}^{*}/E=\{aE:a\in {\U0001d52a}^{*}\}$. Then the subsemigroup $S$ of $K/E$ makes $K/E$ an ordered group equipped with zero. It is not hard to check that the mapping

$$x\mapsto |x|:=xE$$ |

from $K$ to $K/E$ is a Krull valuation of the field $K$.

Title | valuation determined by valuation domain |

Canonical name | ValuationDeterminedByValuationDomain |

Date of creation | 2013-03-22 14:54:58 |

Last modified on | 2013-03-22 14:54:58 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 10 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 13F30 |

Classification | msc 13A18 |

Classification | msc 12J20 |

Classification | msc 11R99 |

Related topic | ValuationDomainIsLocal |

Related topic | KrullValuationDomain |

Related topic | PlaceOfField |