Unique factorization of ideals in a Dedekind domain

Every nonzero proper ideal of a Dedekind domain has a unique factorization, up to reordering, as a finite product of positive powers of distinct nonzero prime ideals. The unit ideal is the empty product. This is distinct from unique factorization of elements.

For a nonzero ideal $I$ in a Dedekind domain, $v_P(I)$ is the exponent of the nonzero prime ideal $P$ in the unique factorization of ideals in a Dedekind domain. Then $I\subseteq J$ if and only if $v_P(I)\ge v_P(J)$ for every $P$, and $v_P(I+J)=\min (v_P(I),v_P(J))$.