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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 .


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.


  • 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.


  • 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).