Discrete valuation ring
|
|
In mathematics, a discrete valuation ring (DVR) is a particular kind of commutative ring that is a local ring, which satisfies conditions that in algebraic geometry come from non-singularity of a point on an algebraic curve. There are many examples that are not geometric in nature.
Formally, 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 noetherian local ring with positive Krull dimension, and the maximal ideal of R is principal.
- R is a local Dedekind domain.
- R is a unique factorization domain with a unique irreducible element (up to units).
- R is local, not a field, and every nonzero fractional ideal of R is irreducible.
- There is some Dedekind valuation ν on the field of fractions K of R, such that R={x:x in K, ν(x) ≥ 0}.
Example
Let R={ p/q : p, q in Z, q odd }. Then the field of fractions of R is Q. Now, for any nonzero element r of Q, we can apply unique factorization to the numerator and denominator of r to write r as 2kp/q, where p, q, and k are integers with p and q odd. In this case, we define ν(r)=k. Then R is the discrete valuation ring corresponding to ν.
Note that R is the localization of the Dedekind domain Z at the prime ideal generated by 2. Any localization of a Dedekind domain is a discrete valuation ring; in practice, this is frequently how discrete valuation rings arise.
References
Atiyah, M. and I. MacDonald. Introduction to Commutative Algebra (1994)
Dummit, David and Richard Foote. Abstract Algebra (2003)