I'm reading about Dedekind domains from Serre's book, and on Pg. 12, before stating Proposition 7, there are arguments for the proof, which read as follows:
If one considers the ideal $a_1 = \displaystyle\prod_{p} p^{\nu_p(a)}$ and the ideal $a_2$ of those $x$ such that $\nu_p(x) \geq \nu_p(a)$ for all $p$, prime ideals in a Dedekind domain A, then the three ideals $a, a_1, a_2$ are equal locally (i.e, they have same images in all $A_p$). Here, $a$ is an arbitrary fractional ideal of $A$.
I don't understand how $a, a_1, a_2$ are equal locally. Please help me to understand.