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 $$IJ=R$$.