Let $\Gamma$ be an abelian group of order $N$. Let $A$ be the maximal order in the ring $\mathbb{Q}[\Gamma]/\sum_{g \in \Gamma}g$. Then in page $53$, section $8$ of this paper the authors say that " It is well know that $A$ is unique and it is a product of ring of integers of number fields." The authors also gives some examples below which are the following:
$1)$ For $\Gamma=\mathbb{Z}/4\mathbb{Z}$, then $A=\mathbb{Z}[i] \times \mathbb{Z}$.
$2)$ For $\Gamma=\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$, then $A=\mathbb{Z} \times\mathbb{Z}\times \mathbb{Z}$.
But I do not understand the examples and the fact that $A$ is unique. Since the authors say that it is well known, could anyone provide me with a reference? Thank you in advance.
This is a short sketch of an explanation; if any part needs expanding let me know.
Because $\Gamma$ is a finite abelian group, the ring $R:=\Bbb{Q}[\Gamma]/\sum_{g\in \Gamma}g$ is a reduced commutative ring that is also a finite dimensional $\Bbb{Q}$-vector space. That means $R$ is Artinian, and hence it has only finitely many prime ideals, and they are all maximal. Then by the Chinese remainder theorem the map $$R\ \longrightarrow\ \prod_{\mathfrak{m}\subset R\text{ maximal}}R/\mathfrak{m},$$ is an isomorphism because $R$ is reduced, where the right hand side is a product of number fields. The maximal order in a number field is its ring of integers, and so the maximal order in $R$ is the product of these rings of integers.