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 which satisfies any one of the following equivalent conditions: R is a local principal ideal domain, and not a field. R is a valuation ring with a value group isomorphic to the integers under addition. R is a local Dedekind domain and not a field. R is a Noetherian local domain whose maximal ideal is principal, and not a field. R is an integrally closed Noetherian local ring with Krull dimension one. R is a principal ideal domain with a unique non-zero prime ideal. R is a principal ideal domain with a unique irreducible element (up to multiplication by units). R is a unique factorization domain with a unique irreducible element (up to multiplication by units). 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. There is some discrete valuation ν on the field of fractions K of R such that R = {0} {x K : ν(x) ≥ 0}. Let . Then, the field of fractions of is . For any nonzero element of , we can apply unique factorization to the numerator and denominator of r to write r as 2k z/n where z, n, and k are integers with z and n odd. In this case, we define ν(r)=k. Then is the discrete valuation ring corresponding to ν. The maximal ideal of is the principal ideal generated by 2, i.e. , and the "unique" irreducible element (up to units) is 2 (this is also known as a uniformizing parameter). Note that is the localization of the Dedekind domain at the prime ideal generated by 2. 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 p in complete analogy. The ring of p-adic integers is a DVR, for any prime .