Discreet evaluation ring
In the mathematical subfield of commutative algebra , discrete evaluation rings are special local rings with particularly good properties.
definition
A discrete evaluation ring is a main local ideal ring that is not a body .
A producer of the maximal ideal is uniformisierendes element or shortly Uniformisierendes . It is also known as DVR (for discrete valuation ring ) or DBR.
properties
- A discrete evaluation ring is a dedekind ring, in particular a regular local integrity ring.
- The spectrum Spec of a discrete evaluation ring consists of exactly two points:
- A closed point, the special point , belonging to the maximum ideal (if the uniformizing element is)
- and an uncompleted (but open) point, the generic point .
- For a review discrete ring is a discrete evaluation on the quotient field defined (if for in ). This rating has as a rating ring.
- If one assigns its evaluation ring to a discretely evaluated body and applies the above construction to it, one obtains a discretely evaluated body that is isomorphic . In other words: these constructions induce an equivalence of categories between discrete valued bodies and discrete valuation rings .
Examples
- The ring of p-adic integers for each prime number . is tight in .
- The ring of rational numbers that are p -adic integer for a prime number . It is and is tight in .
- The ring of formal power series in an indeterminate over a body .
- The ring of convergent power series
- The local ring to a smooth point on an algebraic curve .
literature
- MF Atiyah and IG MacDonald: Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics, 1969, Chapter 9, ISBN 0-201-00361-9