ideal decomposition in Dedekind domain

According to the entry “fractional idealMathworldPlanetmathPlanetmath (”, we can that in a Dedekind domainMathworldPlanetmath R, each non-zero integral ideal 𝔞 may be written as a product of finitely many prime idealsPlanetmathPlanetmathPlanetmath 𝔭i of R,


The product decomposition is unique up to the order of the factors.  This is stated and proved, with more general assumptions, in the entry “prime ideal factorisation is unique (”.

Corollary.  If α1, α2, …, αm are elements of a Dedekind domain R and n is a positive integer, then one has

(α1,α2,,αm)n=(α1n,α2n,,αmn) (1)

for the ideals of R.

This corollary may be proven by induction on the number m of the n).

Title ideal decomposition in Dedekind domain
Canonical name IdealDecompositionInDedekindDomain
Date of creation 2015-05-05 19:05:43
Last modified on 2015-05-05 19:05:43
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 13
Author pahio (2872)
Entry type Topic
Classification msc 11R37
Classification msc 11R04
Related topic ProductOfFinitelyGeneratedIdeals
Related topic PolynomialCongruence
Related topic CancellationIdeal
Related topic DivisibilityInRings
Related topic IdealsOfADiscreteValuationRingArePowersOfItsMaximalIdeal
Related topic DivisorTheory
Related topic GreatestCommonDivisorOfSeveralIntegers