We know that DVRs are Dedekind domains by the characterization of Dedekind domains in terms of localizations at maximal ideals. There are many nice abstract ways to prove that DVRs are Dedekind domains. This is not what this question is about. I would like to see the following:
Can we see explicitly how the existence of a valuation map into $\mathbb Z$ ensures our ring is a Dedekind domain?
Here are the definitions I am using:
$A$ is a dedekind domain if every non zero ideal $\mathfrak{a}$ has a factorization into primary ideals, $\mathfrak{a}=\prod q_i$ where the $q_i$.
$A$ is DVR if it is a valuation ring and the valuation map has image in $\mathbb Z$.
How can we see that every non zero ideal has a factorization into primary ideals using just that we have a valuation map in $\mathbb Z$.