# Dedekindring

A **Dedekind ****ring** (after Richard Dedekind , also **Dedekind area** or **ZPI ring** ) is a generalization of the ring of whole numbers . The applications of this term can be found mainly in the mathematical sub-areas of algebraic number theory and commutative algebra , especially in ideal theory .

## definition

A **Dedekind **ring is at most a one-dimensional , Noetherian , normal integrity ring .

Some authors claim that Dedekind rings are one-dimensional, which means that bodies are not Dedekind rings by definition. However, this is not common.

## properties

- Analogous to the unique decomposition of integers into prime numbers applies to Dedekind domain that in them every ideal a unique
has. Dedekind rings are precisely those integrity**Z**nstallation in**P**rim**i**deale**rings that**are**ZPI rings**.

- Zero-dimensional Dedekind rings are bodies.

- One-dimensional local Dedekind rings are precisely the discrete evaluation rings .

- Every broken ideal different from the zero ideal can be inverted via a Dedekind ring.

- Factorial Dedekind rings are main ideal rings . Conversely, every main ideal ring is a factorial Dedekind ring.

## Examples

- Every main ideal ring (and therefore every discrete evaluation ring ) is a Dedekind ring .
- If a main ideal ring is a finite extension of its quotient field , the whole ending of in is a Dedekind ring . This applies in particular to whole rings in number fields , for example
- Localizations of Dedekind rings are again Dedekind rings.

No Dedekind rings are:

- (two-dimensional),
- (not normal),
- and (no integrity rings),
- the ring of algebraic integers, ie the whole closure of in an algebraic closure of the rational numbers (not noetherian).

## literature

- LA Bokut ': Dedekind ring . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).