Theorem. Every ideal of a Dedekind domain can be generated by two of its elements.
Proof. Let be an arbitrary ideal of a Dedekind domain . Let be such an ideal of that is a principal ideal . The lemma to which this entry is attached gives also an element and an ideal of such that and . Then we have
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 ideal of the Prüfer domain where the ’s run all valuation rings of the rational function field which have the residue fields formally real.
- 1 Eben Matlis: “The two-generator problem for ideals”. – The Michigan Mathematical Journal 17 3 (1970).
- 2 Heinz-Werner Schülting: “Über die Erzeugendenanzahl invertierbarer Ideale in Prüferringen”. – Communications in Algebra 7 13 (1979). [Zentralblatt 432.13010]