two-generator property

Theorem.  Every ideal of a Dedekind domain 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 ideal $(\beta )$.  The lemma to which this entry is attached gives also an element $\gamma$ and an ideal $𝔠$ of $R$ such that  $𝔞𝔠=(\gamma )$  and  $𝔟+𝔠=R$.  Then we have

$$𝔞=\gcd (𝔞𝔟,𝔞𝔠)=\gcd ((\beta ),(\gamma ))=(\beta ,\gamma )$$

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

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  $ℝ(X,Y)$  which have the residue fields formally real.