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
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