If $R$ is a Dedekind domain and $I\subset R$ is a non-zero ideal then by the Noetherian property of $R$, I can show that there are distinct non-zero prime ideals $P_1,...,P_r$ s.t. $P_1^{a_1}\cdots P_r^{a_r}\subset I$ with $a_1,...,a_r\geq 0$.
I'd be grateful for an answer to the following:
If $P$ is a prime ideal in $R$ s.t. $P\notin \{P_1,...,P_r\}$ and $I\subset P$ then is there some $P_i\subset P$?
You can show that the ordinary definition of primes in commutative rings is equivalent to this one:
$P$ is prime if whenever $A,B$ are ideals such that $AB\subseteq P$, then $A\subseteq P$ or $B\subseteq P$.
By applying this to the product of powers if primes in your factorization, you get an affirmative proof.