I have a finite Galois extension $L/K$ of number fields with group $G$. Let $\mathcal O_L$ be the ring of algebraic integers of $L$. We let $\mathcal A_{L/K}$ be the subring $\{x\in K[G]:x\mathcal O_L\subseteq \mathcal O_L\}$ which I think is also a $\mathbb Z$-order in $K[G]$.
My question is if $\mathcal O_L$ is free as an $\mathcal A_{L/K}$-module then is it of rank 1 ?
As ${\cal O}_K[G]\subseteq{\cal A}_{L/K}$ and ${\cal O}_K[G]$ has rank $|G|[K:\Bbb Q]=[L:\Bbb Q]$ as a $\Bbb Z$-module, ${\cal A}_{L/K}$ has at least this rank also as a $\Bbb Z$-module. If ${\cal O}_L$ were a free ${\cal A}_{L/K}$-module of rank $\ge2$, it would necessarily have rank $\ge2[L:\Bbb Q]$ as a $\Bbb Z$-module, but we know ${\cal O}_L$ has rank $[L:\Bbb Q]$ as a $\Bbb Z$-module.