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 .
Some authors claim that Dedekind rings are one-dimensional, which means that bodies are not Dedekind rings by definition. However, this is not common.
- Analogous to the unique decomposition of integers into prime numbers applies to Dedekind domain that in them every ideal a unique Z nstallation in P rim i deale has. Dedekind rings are precisely those integrity rings that are ZPI rings .
- Zero-dimensional Dedekind rings are bodies.
- One-dimensional local Dedekind rings are precisely the discrete evaluation rings .
- Factorial Dedekind rings are main ideal rings . Conversely, every main ideal ring is a factorial Dedekind ring.
- 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:
- (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).