Discrete valuation ring

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:

1. ${\displaystyle R}$ is a local ring, a principal ideal domain, and not a field.
2. ${\displaystyle R}$ is a valuation ring with a value group isomorphic to the integers under addition.
3. ${\displaystyle R}$ is a local ring, a Dedekind domain, and not a field.
4. ${\displaystyle R}$ is Noetherian and a local domain whose unique maximal ideal is principal, and not a field.
5. ${\displaystyle R}$ is integrally closed, Noetherian, and a local ring with Krull dimension one.
6. ${\displaystyle R}$ is a principal ideal domain with a unique non-zero prime ideal.
7. ${\displaystyle R}$ is a principal ideal domain with a unique irreducible element (up to multiplication by units).
8. ${\displaystyle R}$ is a unique factorization domain with a unique irreducible element (up to multiplication by units).
9. ${\displaystyle R}$ is Noetherian, not a field, and every nonzero fractional ideal of R is irreducible in the sense that it cannot be written as a finite intersection of fractional ideals properly containing it.
10. There is some discrete valuation ${\displaystyle \nu }$ on the field of fractions ${\displaystyle K}$ of ${\displaystyle R}$ such that ${\displaystyle R=\{0\}\cup \{x\in K:\nu (x)\geq 0\}}$.