two-generator property

Theorem.  Every ideal of a Dedekind domainMathworldPlanetmath can be generated by two of its elements.

Proof.  Let 𝔞 be an arbitrary ideal of a Dedekind domain R.  Let 𝔟 be such an ideal of R that 𝔞𝔟 is a principal idealMathworldPlanetmath (β).  The lemma to which this entry is attached gives also an element γ and an ideal 𝔠 of R such that  𝔞𝔠=(γ)  and  𝔟+𝔠=R.  Then we have


because  gcd(𝔟,𝔠)=𝔟+𝔠=R=(1).

The Dedekind domains are trivially Prüfer domains, but the two-generator property can not be generalized to the invertible ideals of all Prüfer domains (and Prüfer rings):  Schülting has constructed an invertible ideal of a Prüfer domain that can not be generated by less than three generators.  The example of Schülting is the fractional idealMathworldPlanetmath(1,X,Y)  of the Prüfer domain  jBj  where the Bj’s run all valuation ringsMathworldPlanetmathPlanetmath of the rational function fieldPlanetmathPlanetmath(X,Y)  which have the residue fieldsMathworldPlanetmath formally real.


  • 1 Eben Matlis: “The two-generator problem for ideals”.  – The Michigan Mathematical Journal 17N 3 (1970).
  • 2 Heinz-Werner Schülting: “Über die Erzeugendenanzahl invertierbarer Ideale in Prüferringen”.  – Communications in Algebra 7N 13 (1979). [Zentralblatt 432.13010]
Title two-generator property
Canonical name TwogeneratorProperty
Date of creation 2015-05-05 15:25:37
Last modified on 2015-05-05 15:25:37
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 38
Author pahio (2872)
Entry type Theorem
Classification msc 11R04
Synonym Ideal of Dedekind domain
Related topic SumOfIdeals
Related topic FamousAndInfamousOpenQuestionsInMathematics
Related topic AnyDivisorIsGcdOfTwoPrincipalDivisors