proof that a domain is Dedekind if its ideals are products of maximals
Let R be an integral domain. We show that it is a Dedekind domain if and only if every nonzero proper ideal
can be expressed as a product
of maximal ideals
.
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 x∈𝔪. Then, by assumption,
(x)=𝔪1⋯𝔪𝔫 |
for some n≥0 and maximal ideals 𝔪k. As (x) is a principal ideal, each of the factors 𝔪k is invertible.
Also,
𝔪1⋯𝔪𝔫⊆𝔪. |
As 𝔪 is prime, this gives 𝔪k⊆𝔪 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 contradiction 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 Noetherian
(http://planetmath.org/Noetherian), there is an ideal 𝔞 maximal (http://planetmath.org/MaximalElement) (w.r.t. the partial order
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 𝔞⊆𝔟 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 |