# two-generator property

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

Proof.  Let $\mathfrak{a}$ be an arbitrary ideal of a Dedekind domain $R$.  Let $\mathfrak{b}$ be such an ideal of $R$ that $\mathfrak{ab}$ is a principal ideal $(\beta)$.  The lemma to which this entry is attached gives also an element $\gamma$ and an ideal $\mathfrak{c}$ of $R$ such that  $\mathfrak{ac}=(\gamma)$  and  $\mathfrak{b+c}=R$.  Then we have

 $\mathfrak{a}=\gcd(\mathfrak{ab},\,\mathfrak{ac})=\gcd((\beta),\,(\gamma))=(% \beta,\,\gamma)$

because  $\gcd(\mathfrak{b},\,\mathfrak{c})=\mathfrak{b+c}=R=(1)$. $\Box$

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$(1,\,X,\,Y)$  of the Prüfer domain  $\bigcap_{j}B_{j}$  where the $B_{j}$’s run all valuation rings of the rational function field$\mathbb{R}(X,\,Y)$  which have the residue fields formally real.

## References

• 1 Eben Matlis: “The two-generator problem for ideals”.  – The Michigan Mathematical Journal 17$\mbox{N}^{\circ}$ 3 (1970).
• 2 Heinz-Werner Schülting: “Über die Erzeugendenanzahl invertierbarer Ideale in Prüferringen”.  – Communications in Algebra 7$\mbox{N}^{\circ}$ 13 (1979). [Zentralblatt 432.13010]
Title two-generator property TwogeneratorProperty 2015-05-05 15:25:37 2015-05-05 15:25:37 pahio (2872) pahio (2872) 38 pahio (2872) Theorem msc 11R04 Ideal of Dedekind domain SumOfIdeals FamousAndInfamousOpenQuestionsInMathematics AnyDivisorIsGcdOfTwoPrincipalDivisors