Let $K$ be an algebraic number field and $\mathcal{O}_K$ its ring of integers. For a non-zero ideal $\mathfrak{a}$ of $\mathcal{O}_K$ and an element $c \in \mathcal{O}_K \setminus \{0\}$ I wonder whether we always have an isomorphism $$ \mathfrak{a} / c \mathfrak{a} \cong \mathcal{O}_K / (c) $$ as $\mathcal{O}_K$-modules.
Using the inverse (fractional) ideal $\mathfrak{a}^{-1}$, one could naïvely "calculate" $$ \mathfrak{a} / (c) \mathfrak{a} \cong \mathfrak{a}\mathfrak{a}^{-1} / (c)\mathfrak{a}\mathfrak{a}^{-1} = \mathcal{O}_K / (c)\mathcal{O}_K = \mathcal{O}_K / (c). $$ But (again) I do not know how to justify the isomorphism.
Additonal question
Is it easier to somehow only show $ [\mathfrak{a} : c \mathfrak{a}] = [\mathcal{O}_K : (c)]$ ? This would help me, too :-).