proof that a domain is Dedekind if its ideals are products of maximals

Let R be an integral domainMathworldPlanetmath. We show that it is a Dedekind domain if and only if every nonzero proper idealMathworldPlanetmathPlanetmath can be expressed as a productPlanetmathPlanetmathPlanetmath of maximal idealsMathworldPlanetmath. To do this, we make use of the characterization of Dedekind domains as integral domains in which every nonzero integral ideal is invertible (proof that a domain is Dedekind if its ideals are invertible).

First, let us suppose that every nonzero proper ideal in R is a product of maximal ideals. Let 𝔪 be a maximal ideal and choose a nonzero x𝔪. Then, by assumptionPlanetmathPlanetmath,


for some n0 and maximal ideals 𝔪k. As (x) is a principal idealMathworldPlanetmathPlanetmath, each of the factors 𝔪k is invertible. Also,


As 𝔪 is prime, this gives 𝔪k𝔪 for some k. However, 𝔪k is maximal so must equal 𝔪, showing that 𝔪 is indeed invertible. Then, every nonzero proper ideal is a product of maximal, and hence invertible, ideals and so is invertible, and it follows that R is Dedekind.

We now show the reverse direction, so suppose that R is Dedekind. Proof by contradictionMathworldPlanetmathPlanetmath will be used to show that every nonzero ideal is a product of maximals, so suppose that this is not the case. Then, as R is defined to be NoetherianPlanetmathPlanetmathPlanetmath (, there is an ideal 𝔞 maximal ( (w.r.t. the partial orderMathworldPlanetmath of set inclusion) among those proper ideals which are not a product of maximal ideals. Then 𝔞 cannot be a maximal ideal itself, so is strictly contained in a maximal ideal 𝔪 and, as 𝔪 is invertible, we can write 𝔞=𝔪𝔟 for an ideal 𝔟.

Therefore 𝔞𝔟 and we cannot have equality, otherwise cancelling 𝔞 from 𝔞=𝔪𝔞 would give 𝔪=R. So, 𝔟 is strictly larger than 𝔞 and, by the choice of 𝔞, is therefore a product of maximal ideals. Finally, 𝔞=𝔪𝔟 is then also a product of maximal ideals.

Title proof that a domain is Dedekind if its ideals are products of maximals
Canonical name ProofThatADomainIsDedekindIfItsIdealsAreProductsOfMaximals
Date of creation 2013-03-22 18:35:04
Last modified on 2013-03-22 18:35:04
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Proof
Classification msc 13F05
Classification msc 13A15
Related topic DedekindDomain
Related topic MaximalIdeal
Related topic FractionalIdeal