Dedekind domain

plural Dedekind domains

1. ring theoryalgebraAn 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.

   3 more examples

   In this chapter we shall study several of the important classes of rings which contain the class of Dedekind domains.

   1971, Max D. Larsen, Paul J. McCarthy, Multiplicative Theory of Ideals, Elsevier (Academic Press), page 201

   As we can see every principal ideal domain is a Dedekind domain.

   2007, Leonid Kurdachenko, Javier Otal, Igor Ya. Subbotin, Artinian Modules over Group Rings, Springer (Birkhäuser), page 55

   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.

   2007, Anthony W. Knapp, Advanced Algebra, Springer (Birkhäuser), page 266

   Synonyms: Dedekind ring