Two generators for an ideal in a Dedekind domain

Every ideal of a Dedekind domain is generated by at most two elements. Given a nonzero ideal $I$ and any prescribed nonzero $a\in I$, the Chinese remainder theorem supplies $b\in I$ with $v_P((b))=v_P(I)$ at every prime ideal dividing $(a)$. The prime-ideal valuation in a Dedekind domain then gives $I=(a,b)$.