Dedekind domain

A Dedekind ring ( after Richard Dedekind, also Dedekind area or ZPI -ring) is a generalization of the ring of integers. The applications of this concept are found primarily in the mathematical sciences in algebraic number theory and commutative algebra, especially in the theory of ideals.

Definition

A Dedekind ring is a maximum of one-dimensional Noetherian, normal integral domain.

Properties

Analogous to the unique decomposition of integers into primes is true for Dedekind rings in them that every ideal has a unique decomposition into prime ideals. Dedekind rings are precisely those integrity rings ZPI -rings.

Examples

  • Every principal ideal (and hence any discrete valuation ring ) is a Dedekind ring.
  • Is a principal ideal domain, and a finite extension of its quotient field, then the whole concluding in a Dedekind ring. In particular, this applies to wholeness rings in number fields, ie for example
  • Localizations of Dedekind rings are Dedekind rings again.

No Dedekind rings are:

  • ( two-dimensional),
  • (not normal),
  • And ( no integrity rings).
Mathematics Height (ring theory) Michiel Hazewinkel Encyclopedia of Mathematics
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