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 R that satisfies any and all of the following equivalent conditions:

1. R is a local ring, a principal ideal domain, and not a field.
2. R is a valuation ring with a value group isomorphic to the integers under addition.
3. R is a local ring, a Dedekind domain, and not a field.
4. R is Noetherian and a local domain whose unique maximal ideal is principal, and not a field.
5. R is integrally closed, Noetherian, and a local ring with Krull dimension one.
6. R is a principal ideal domain with a unique non-zero prime ideal.
7. R is a principal ideal domain with a unique irreducible element (up to multiplication by units).
8. R is a unique factorization domain with a unique irreducible element (up to multiplication by units).
9. 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 ν on the field of fractions K of R such that R = {0} ${\displaystyle \cup }$ {x ${\displaystyle \in }$ K : ν(x) ≥ 0}.