Dedekind domain

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English

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Etymology

Named after German mathematician Richard Dedekind (1831–1916).

Noun

Dedekind domain (plural Dedekind domains)

  1. (algebra, ring theory) An integral domain in which every proper ideal factors into a product of prime ideals which is unique (up to permutations).
    It can be proved that a Dedekind domain (as defined above) is equivalent to an integral domain in which every proper fractional ideal is invertible.
    • 1971, Max D. Larsen, Paul J. McCarthy, Multiplicative Theory of Ideals‎, Elsevier (Academic Press), page 201:
      In this chapter we shall study several of the important classes of rings which contain the class of Dedekind domains.
    • 2007, Leonid Kurdachenko, Javier Otal, Igor Ya. Subbotin, Artinian Modules over Group Rings, Springer (Birkhäuser), page 55:
      As we can see every principal ideal domain is a Dedekind domain.
    • 2007, Anthony W. Knapp, Advanced Algebra‎, Springer (Birkhäuser), page 266:
      Let us recall some material about Dedekind domains from Chapters VIII and IX of Basic Algebra. A Dedekind domain is a Noetherian integral domain that is integrally closed and has the property that every nonzero prime ideal is maximal. Any Dedekind domain has unique factorization for its ideals.

Usage notes

Synonyms

  • (integral domain whose prime ideals factorise uniquely): Dedekind ring

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