# valuation domain is local

###### Theorem.

Every valuation domain is a local ring^{}.

Proof. Let $R$ be a valuation domain and $K$ its field of fractions^{}. We shall show that the set of all non-units of $R$ is the only maximal ideal^{} of $R$.

Let $a$ and $b$ first be such elements of $R$ that $a-b$ is a unit of $R$; we may suppose that $ab\ne 0$ since otherwise one of $a$ and $b$ is instantly stated to be a unit. Because $R$ is a valuation domain in $K$, therefore e.g. $\frac{a}{b}\in R$. Because now $\frac{a-b}{b}=1-\frac{a}{b}$ and ${(a-b)}^{-1}$ belong to $R$, so does also the product $\frac{a-b}{b}\cdot {(a-b)}^{-1}=\frac{1}{b}$, i.e. $b$ is a unit of $R$. We can conclude that the difference $a-b$ must be a non-unit whenever $a$ and $b$ are non-units.

Let $a$ and $b$ then be such elements of $R$ that $ab$ is its unit, i.e. ${a}^{-1}{b}^{-1}\in R$. Now we see that

$${a}^{-1}=b\cdot {a}^{-1}{b}^{-1}\in R,{b}^{-1}=a\cdot {a}^{-1}{b}^{-1}\in R,$$ |

and consequently $a$ and $b$ both are units. So we conclude that the product $ab$ must be a non-unit whenever $a$ is an element of $R$ and $b$ is a non-unit.

Thus the non-units form an ideal $\U0001d52a$. Suppose now that there is another ideal $\U0001d52b$ of $R$ such that $\U0001d52a\subset \U0001d52b\subseteq R$. Since $\U0001d52a$ contains all non-units, we can take a unit $\epsilon $ in $\U0001d52b$. Thus also the product ${\epsilon}^{-1}\epsilon $, i.e. 1, belongs to $\U0001d52b$, or $R\subseteq \U0001d52b$. So we see that $\U0001d52a$ is a maximal ideal. On the other hand, any maximal ideal of $R$ contains no units and hence is contained in $\U0001d52a$; therefore $\U0001d52a$ is the only maximal ideal.

Title | valuation domain is local |
---|---|

Canonical name | ValuationDomainIsLocal |

Date of creation | 2013-03-22 14:54:49 |

Last modified on | 2013-03-22 14:54:49 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 10 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 13F30 |

Classification | msc 13G05 |

Classification | msc 16U10 |

Related topic | ValuationRing |

Related topic | ValuationDeterminedByValuationDomain |

Related topic | HenselianField |