Dedekind domain

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

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.

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

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

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

• almost Dedekind domain

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