Anything wrong with this proof?

Question

So there's a result that goes as follows: if $ A $ is a Dedekind ring and ideals $ \mathfrak{a}, \mathfrak{b} $ satisfy $ \mathfrak{a} \supseteq \mathfrak{b} $ then $ \mathfrak{a} \mid \mathfrak{b} $ as ideals. I've seen a few proofs of this result and it always uses the fractional ideals idea. However, since we know that we have unique factorization into prime ideals, if $ \mathfrak{a} \supseteq \mathfrak{b} $ then we have prime fractorizations $ \mathfrak{a} = \prod_i \mathfrak{p}_i^{r_i} $ and $ \mathfrak{b} = \prod_i \mathfrak{p_i}^{s_i} $ such that $ r_i \leq s_i $. Then letting $ c = \prod_i \mathfrak{p}_i^{s_i - r_i} $ we should have $ \mathfrak{a} \mathfrak{c} = \mathfrak{b} $ right? Why even need fractional ideals?