If $K = \mathbb{Q}[\sqrt{-5}]$ and $\mathfrak{a}=(2,1+\sqrt{-5})$ then the $\mathfrak{a}$ as a module is isomorphic to $2\cdot\mathbb{Z} \oplus (1+\sqrt{-5})\cdot\mathbb{Z} \cong \mathbb{Z}^2$ and the elements in its integral basis are $2$ which is a vector $(2,0)^T$ and $1+\sqrt{-5}$ which is $(1,1)^T$. So basis of $\mathfrak{a}$ is $\begin{pmatrix} 2 & 1\\ 0 & 1 \end{pmatrix}$.

Yes! The $\mathbb{Z}$ is typo. I meant that $\mathfrak{a}$ is isomorphic to $\bigoplus_{i} a_i \cdot \mathbb{Z}$ because it is $\mathcal{O}_K$-module (for $\mathcal{O}_K$ - ring of integers of $K$) and therefore a $\mathbb{Z}$ module itself. I'll send an example in the next comment.