The Stacks project

Definition 10.50.13. Let $A$ be a valuation ring.

  1. The totally ordered abelian group $(\Gamma , \geq )$ of Lemma 10.50.12 is called the value group of the valuation ring $A$.

  2. The map $v : A - \{ 0\} \to \Gamma $ and also $v : K^* \to \Gamma $ is called the valuation associated to $A$.

  3. The valuation ring $A$ is called a discrete valuation ring if $\Gamma \cong \mathbf{Z}$.

Comments (2)

Comment #8751 by Anonymous on

Pedantic point, but do you want to allow fields to be discrete valuation rings? For example, Tag 034X seems to consider fields as discrete valuation rings, but the value group will trivial.

There are also:

  • 4 comment(s) on Section 10.50: Valuation rings

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00IE. Beware of the difference between the letter 'O' and the digit '0'.



View Definition 10.50.13 as pdf