Based on arguments in Marcus’s Number Fields and K. Conrad’s expository paper on ideal factorization. I assume familiarity with the concepts related to ideals and fractional ideals in a commutative ring.
Recall that an integral domain \(R\) with the field of fractions \(K\) is called a Dedekind domain if the following conditions hold:
- \(R\) is Noetherian, so every nonempty set of ideals has a maximal element, or equivalently, every ideal is finitely generated,
- every prime ideal in \(R\) is a maximal ideal, and
- \(R\) is integrally closed in \(K\), so that every root of a monic polynomial from \(R[x]\) lying in \(K\) lies in \(R\).
Our goal is to prove the following proposition:
Proposition: Assume \(R\) is a Dedekind domain. For any nonzero (i.e. containing a nonzero element) fractional ideal \(I\) in \(K\) there exists a fractional ideal \(J\) such that \(IJ=R\).