In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.

This means a DVR is an integral domain ${\displaystyle R}$ that satisfies any and all of the following equivalent conditions:

Let ${\displaystyle \mathbb {Z} _{(2)}}$ be the localization of ${\displaystyle \mathbb {Z} }$ at the ideal generated by 2. Formally,

The field of fractions of ${\displaystyle \mathbb {Z} _{(2)}}$ is ${\displaystyle \mathbb {Q} }$. For any nonzero element ${\displaystyle r}$ of ${\displaystyle \mathbb {Q} }$, we can apply unique factorization to the numerator and denominator of ${\displaystyle r}$ to write ${\displaystyle r}$ as ${\displaystyle {\tfrac {2^{k}z}{n}}}$ where ${\displaystyle z}$, ${\displaystyle n}$, and ${\displaystyle k}$ are integers with ${\displaystyle z}$ and ${\displaystyle n}$ odd. In this case, we define ${\displaystyle \nu (r)=k}$.

Then ${\displaystyle \mathbb {Z} _{(2)}}$ is the discrete valuation ring corresponding to ${\displaystyle \nu }$. The maximal ideal of ${\displaystyle \mathbb {Z} _{(2)}}$ is the principal ideal generated by 2; i.e., ${\displaystyle 2\mathbb {Z} _{(2)}}$, and the "unique" irreducible element (up to units) is 2 (also known as a uniformizing parameter).

More generally, any localization of a Dedekind domain at a non-zero prime ideal is a discrete valuation ring; in practice, this is frequently how discrete valuation rings arise. In particular, we can define rings

for any prime ${\displaystyle p}$ in complete analogy.

The ring ${\displaystyle \mathbb {Z} _{p}}$ of p-adic integers is a DVR, for any prime ${\displaystyle p}$. Here ${\displaystyle p}$ is an irreducible element; the valuation assigns to each ${\displaystyle p}$-adic integer ${\displaystyle x}$ the largest integer ${\displaystyle k}$ such that ${\displaystyle p^{k}}$ divides ${\displaystyle x}$.