Dedekind domain
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English
[edit]Etymology
[edit]Named after German mathematician Richard Dedekind (1831–1916).
Noun
[edit]Dedekind domain (plural Dedekind domains)
- (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[1], 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[2], 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
[edit]- For a list of equivalent definitions, see
Dedekind domain on Wikipedia.Wikipedia
Synonyms
[edit]- (integral domain whose prime ideals factorise uniquely): Dedekind ring
Hypernyms
[edit]- (integral domain whose prime ideals factorise uniquely): Noetherian domain
Derived terms
[edit]Translations
[edit]integral domain whose prime ideals factorise uniquely
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Further reading
[edit]Ideal class group on Wikipedia.Wikipedia
Unique factorization domain on Wikipedia.Wikipedia
- Dedekind ring on Encyclopedia of Mathematics
- Dedekind Ring on Wolfram MathWorld