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]

## Comments

## prufer domain, and two generator propert, unsolved?

Did not Schutling (1979) give an example of a Prufer domain that required 3 generators?

Schutling, ÃƒÂœber die Erzeugendenanzahl invertierbarer Ideale in Pruferringen, Comm. Algebra, 7 (1979), no. 13, 1331-1349.

## Re: prufer domain, and two generator propert, unsolved?

Thank you, nkadambi, very interesting! I must read it.

Regards,

Jussi

## Re: prufer domain, and two generator propert, unsolved?

The two-generator property

http://planetmath/encyclopedia/TwoGeneratorProperty.html

has now been updated (ich las den Artikel Sch\"utlings).

Jussi

## Re: Pruefer domain, and two generator property, solved

The two-generator property

http://planetmath.org/encyclopedia/TwoGeneratorProperty.html

has now been updated (ich las den Artikel Sch\"utlings).

The address improved.

Jussi