# ideal decomposition in Dedekind domain

According to the entry “fractional ideal   (http://planetmath.org/FractionalIdeal)”, we can that in a Dedekind domain  $R$, each non-zero integral ideal $\mathfrak{a}$ may be written as a product of finitely many prime ideals   $\mathfrak{p}_{i}$ of $R$,

 $\mathfrak{a}=\mathfrak{p}_{1}\mathfrak{p}_{2}...\mathfrak{p}_{k}.$

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 (http://planetmath.org/PrimeIdealFactorizationIsUnique)”.

Corollary.  If $\alpha_{1}$, $\alpha_{2}$, …, $\alpha_{m}$ are elements of a Dedekind domain $R$ and $n$ is a positive integer, then one has

 $\displaystyle(\alpha_{1},\,\alpha_{2},\,...,\,\alpha_{m})^{n}=(\alpha_{1}^{n},% \,\alpha_{2}^{n},\,...,\,\alpha_{m}^{n})$ (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