Discrete valuation ring

Let $R$ be a commutative ring.

Definition

A discrete valuation ring (DVR) is an local ring $(R,\mathfrak m)$ such that:

• $R$ is an integral domain ,
• $R$ is Noetherian ,
• the maximal ideal $\mathfrak m$ is principal: $\mathfrak m=(\pi)$ for some $\pi\in R$,
• and $\dim R = 1$ (Krull dimension ).

The generator $\pi$ is called a uniformizer. In a DVR, every nonzero ideal has the form $(\pi^n)$ for a unique $n\ge 0$.

Equivalent characterizations (useful in practice):

• A DVR is exactly a PID with a unique nonzero prime ideal.
• A DVR is exactly a localization at a nonzero prime of a Dedekind domain .

The residue field is $R/\mathfrak m$ (see residue field ).

Examples

1. $p$-local integers.
   $\mathbb{Z}_{(p)}$ (localizing $\mathbb{Z}$ at the prime $(p)$) is a DVR with uniformizer $p$.

2. Formal power series.
   For a field $k$, the ring $k[[t]]$ is a DVR with uniformizer $t$; its maximal ideal is $(t)$.

3. A localized polynomial ring.
   $k[x]_{(x)}$ is a DVR with uniformizer $x$ (it is local with maximal ideal generated by $x$).