Dedekind domain

English

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

• For a list of equivalent definitions, see Dedekind domain#Alternative definitions on Wikipedia.Wikipedia

Synonyms

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

Hypernyms

• (integral domain whose prime ideals factorise uniquely): Noetherian domain

Derived terms

• almost Dedekind domain

Translations

integral domain whose prime ideals factorise uniquely

• French: anneau de Dedekind m
• German: Dedekindring m

• Italian: dominio di Dedekind m, anello di Dedekind m
• Portuguese: domínio de Dedekind m, anel de Dedekind m

Further reading

• Ideal class group on Wikipedia.Wikipedia
• Unique factorization domain on Wikipedia.Wikipedia
• Dedekind ring on Encyclopedia of Mathematics
• Dedekind Ring on Wolfram MathWorld