Discrete valuation ring

In the mathematical field of algebra are discrete valuation rings certain local rings with particularly good properties.

Definition: A discrete valuation ring is a local principal ideal ring which is not a body. A generator of the maximal ideal is called uniformisierendes element or short Uniformisierendes. It also writes short DVR ( discrete valuation ring for ) or DBR.

Properties

• A discrete valuation ring is a Dedekind ring, especially a regular local ring integrity.
• The spectrum of a discrete valuation ring Spec consists of exactly two points:

A closed point, the particular point belonging to the maximal ideal (if the uniformizing element ) and an unfinished (but open ) point, the generic point.

• For a discrete valuation ring is defined by a discrete valuation on the quotient field (if for in ). This review has as a valuation ring.
• If one arranges a discretely valued field to its valuation ring and applies the above construction, so one obtains a discrete valued field, which is isomorphic to. In other words, these structures induce an equivalence of categories between discrete valued bodies and discrete valuation rings.

Examples

• The ring around the p- adic numbers for each prime. is dense in.
• The ring of rational numbers, the p- adic are all, for a prime number. It is and is dense in.
• The ring of formal power series in one variable over a field.
• The ring of convergent power series

• The local ring at a smooth point of an algebraic curve.

• Commutative Algebra