I am struggling with some "usual" maps in algebraic number theory, but with which I am not used to, confusing a lot of different settings, as idèles, corresponding ideals and reduced images modulo something. And since we sometimes barely define precisely the maps we use, I stay confused. I would like at last to be able to understand as explicitly as possible the involved maps (see below).
Let $K$ be a quadratic number field, I want to grasp quite explicitly the following "natural" maps (I am uncertain of what I write, and will be glad to read every corrections or clarifications you can make):
$$\mathrm{Ideals \ of \ } \mathcal{O}_K \to \mathbf{A}_{K,f}^\times/K^\times \to \hat{\mathcal{O}}_K^\times/(1+\mathfrak{m}\hat{\mathcal{O}}_K^\times) \to \mathcal{O}_K/\mathfrak{m} \leftarrow \mathbf{Z}/\mathfrak{m}\cap\mathbf{Z}$$
Let us be more precise from now on. Take $\mathfrak{a}$ to be an ideal of $\mathcal{O}_K$, the ring of integers of $K$. Write its decomposition into prime ideals:
$$\mathfrak{a} = \prod_i \mathfrak{p}_i^{e_i} \quad \mathrm{\ an \ ideal \ of \ } \mathcal{O}_K$$
We can embed it in the idèles on $K$. For that, I choose local uniformizers $\varpi_{\mathfrak{p}_i}$ in $K_{\mathfrak{p}_i}$ for each $i$, and consider the (is it canonical? could we come back to $\mathfrak{a}$?) image:
$$x = \prod_i \varpi_{\mathfrak{p}_i}^{e_i} \quad \in \mathbf{A}_{K,f}^\times/K^\times$$
Now, we want to "reduce it modulo an ideal $\mathfrak{m}$" prime to $\mathfrak{a}$, that is to say we consider its image modulo $1+\mathfrak{m}\hat{\mathcal{O}}_K^\times$. But here I get lost and do not understand what "reducing modulo $\mathfrak{m}$ can mean for these ideles...:
$$\overline{x} = \prod_i \overline{\varpi_{\mathfrak{p}_i}}^{e_i} \quad \in \hat{\mathcal{O}}_K^\times / (1+\mathfrak{m}\hat{\mathcal{O}}_K^\times) \cong \mathcal{O}_K/\mathfrak{m}$$
Back to $\mathcal{O}_K$, denote $x_0$ the image in the last quotient. Now come my embarrassment: there is a natural map $\mathbf{Z}/\mathfrak{m}\cap\mathbf{Z} \to \mathcal{O}_K/\mathfrak{m}$ coming from $1 \mapsto 1_K$ (that is to say just viewing $\mathcal{O}_K/\mathfrak{m}$ as a $\mathbf{Z}$-module), and here is my question:
I would like to determine whether or not $x_0$ is in its image, depending on $\mathfrak{a}$, i.e. what can be said about $\mathfrak{a}$, the $\mathfrak{p}_i$ and/or the $e_i$ for having $x_0$ in $\mathbf{Z}/\mathfrak{m}\cap\mathbf{Z}$?
If necessary we can suppose $\mathfrak{m}$ is a power of prime, by Chinese remainder theorem.
Any idea or reference will be really welcome.
(My question could seem in some ways naive or elementary, so I wonder if this question is more suited for MO or MSE, hence I apologize in advance if I am wrong.)