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