proof that a domain is Dedekind if its ideals are invertible

Let R be an integral domainMathworldPlanetmath with field of fractionsMathworldPlanetmath k. We show that the following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath.

  1. 1.

    R is Dedekind. That is, it is NoetherianPlanetmathPlanetmathPlanetmath (, integrally closedMathworldPlanetmath, and every prime idealMathworldPlanetmathPlanetmathPlanetmath is maximal (

  2. 2.

    Every nonzero (integral) ideal is invertiblePlanetmathPlanetmath.

  3. 3.

    Every fractional ideal is invertible.

As every fractional ideal is the productPlanetmathPlanetmath of an element of k and an integral ideal, statements (2) and (3) are equivalent. We start by proving that (3) implies R is Dedekind.


If every fractional ideal is invertible, then R is Dedekind.


First, every invertible ideal is finitely generated, so R is Noetherian.

Now, let 𝔭 be a prime ideal, and 𝔪 be a maximal idealMathworldPlanetmath containing 𝔭. As 𝔪 is invertible, there exists an ideal 𝔞 such that 𝔭=𝔪𝔞. That 𝔭 is a prime ideal implies 𝔞𝔭 or 𝔪𝔭. The first case gives 𝔭𝔪𝔭 and, by cancelling the invertible ideal 𝔭 implies that 𝔪=R, a contradictionMathworldPlanetmathPlanetmath. So, the second case must be true and, by maximality of 𝔪, 𝔭=𝔪, showing that all prime ideals are maximal.

Now let x be an element of the field of fractions k and be integral over R. Then, we can write xn=c0+c1x++cn-1xn-1 for coefficients ckR. Letting 𝔞 be the fractional ideal


gives xn𝔞, so x𝔞𝔞. As 𝔞 is invertible, it can be cancelled to give xR, showing that R is integrally closed. ∎

It only remains to show the converseMathworldPlanetmath, that is if R is Dedekind then every nonzero ideal is invertible. We start with the following lemmas.


Every nonzero ideal a contains a product of prime ideals. That is, p1pna for some nonzero prime ideals pk.


We use proof by contradictionMathworldPlanetmath, so suppose this is not the case. As R is Noetherian, the set of nonzero ideals which do not contain a product of nonzero primes has a maximal elementMathworldPlanetmath (w.r.t. the partial orderMathworldPlanetmath of set inclusion) say, 𝔞.

In particular 𝔞 cannot be prime itself, so there exist x,yR such that xy𝔞 and x,y𝔞. Therefore 𝔞 is strictly contained in 𝔞+(x) and 𝔞+(y) and, by the choice of 𝔞, these ideals must contain a product of primes. So,


contains a product of primes, which is the required contradiction. ∎


For any nonzero proper idealMathworldPlanetmath a there is an element xkR such that xaR.


Let 𝔭 be a maximal ideal containing 𝔞 and a be a nonzero element of 𝔞. By the previous lemma there are prime ideals 𝔭1,,𝔭n satisfying


We choose n as small as possible. As 𝔭 is prime, this gives 𝔭k𝔭 for some k and, as every prime ideal is maximal, this is an equality. Without loss of generality we may take 𝔭=𝔭n. As n was assumed to be as small as possible, 𝔭1𝔭n-1 is not a subset of (a), so there exists b𝔭1𝔭n-1(a). Then, b(a) gives xa-1bR and


as required. ∎

We finally show that every nonzero ideal 𝔞 is invertible. If its inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath exists then it should be the largest fractional ideal satisfying 𝔟𝔞R, so we set


Choosing any nonzero a𝔞 gives a𝔟𝔟𝔞R so 𝔟 is indeed a fractional ideal. It only remains to be shown that 𝔟𝔞=R, for which we use proof by contradiction. If this were not the case then the previous lemma gives an xkR such that x𝔟𝔞R. By the definition of 𝔟, this gives x𝔟𝔟 and therefore 𝔟 is an R[x]-module. Furthermore, as R is Noetherian, 𝔟 will be finitely generatedMathworldPlanetmathPlanetmath as an R-module. This implies that x is integral over the integrally closed ring R, so xR, giving the required contradiction.

Title proof that a domain is Dedekind if its ideals are invertible
Canonical name ProofThatADomainIsDedekindIfItsIdealsAreInvertible
Date of creation 2013-03-22 18:34:54
Last modified on 2013-03-22 18:34:54
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Proof
Classification msc 13A15
Classification msc 13F05
Related topic DedekindDomain
Related topic FractionalIdeal