splitting the sum into two sums

Question

Why can I split the sum $$\sum_{k\equiv0(5)}e^{2\pi i k /5}$$ into $$\sum_{k\ mod\ 10 \\k \equiv0(5)}\sum_{r\equiv k(10)}e^{2\pi ir/5}?$$ Actually I am trying to understand why the sum $\sum_{\nu\equiv0(\mathfrak{\overline a})}$ can be split into $$\sum_{\alpha\ mod (\mathfrak{\overline a}Q\sqrt{D})\\\alpha \equiv0(\mathfrak{ \overline a})}\sum_{\mu\equiv\alpha(\mathfrak{\overline a}Q\sqrt{D})} ,$$ where $\mathfrak{a}$ is an ideal with its conjugate ideal $\mathfrak{ \overline a}, Q \in \mathbb{N},D \in \mathbb{Z}.$