## You are here

Homevaluation domain is local

## Primary tabs

# 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 $ab\neq 0$ 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 $\mathfrak{m}$. Suppose now that there is another ideal $\mathfrak{n}$ of $R$ such that $\mathfrak{m}\subset\mathfrak{n}\subseteq R$. Since $\mathfrak{m}$ contains all non-units, we can take a unit $\varepsilon$ in $\mathfrak{n}$. Thus also the product $\varepsilon^{{-1}}\varepsilon$, i.e. 1, belongs to $\mathfrak{n}$, or $R\subseteq\mathfrak{n}$. So we see that $\mathfrak{m}$ is a maximal ideal. On the other hand, any maximal ideal of $R$ contains no units and hence is contained in $\mathfrak{m}$; therefore $\mathfrak{m}$ is the only maximal ideal.

## Mathematics Subject Classification

13F30*no label found*13G05

*no label found*16U10

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections