= Solution
{c}
The <unique factorization of ideals in a Dedekind domain> states that every nonzero proper <ideal> $I$ of a <Dedekind domain> $D$ has a factorization
$$
\boxed{I=P_1^{n_1}\cdots P_r^{n_r},}
$$
where the $P_j$ are distinct nonzero <prime ideals> and the $n_j$ are positive integers. The <prime ideals> and their exponents are unique up to reordering. The unit <ideal> $D$ is the empty product; the zero <ideal> is excluded.
Write $v_P(I)$ for the <prime-ideal valuation in a Dedekind domain>, namely the exponent of $P$ in this factorization, with zero assigned when $P$ is absent. The associated containment rule is
$$
I\subseteq J\quad\Longleftrightarrow\quad v_P(I)\geq v_P(J)\text{ for every nonzero prime ideal }P.
$$
Larger exponents give smaller <ideals>, just as higher divisibility gives smaller <principal ideals> in $\mathbb Z$. The theorem concerns unique factorization of <ideals>; it does not assert unique factorization of elements or that these <ideals> are principal.
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