Let $S$ be a ring and take a finite group $G$ acting on $S$. Put $R = S^G$, the fixed subring. When is $S$ a free rank-$1$ $R[G]$-module? That is, when do we have $S \cong R[G]$ as $R[G]$-modules?
I am sure this is known, and I am aware of the normal basis theorem for the case of fields, but this is more general. I am interested in a complete characterization of when this proeprty holds, or any partial results that do not merely apply to fields.
For one example, I conjecture that this holds for integrally closed domains. For an extension of number fields $\mathcal{O}_L / \mathcal{O}_K$ with $L/K$ Galois, I would suspect this to follow from the normal basis theorem.
Edit: my guess here was shown to be false in the comments.
Does anyone know any cool results on the matter more general than the normal basis theorem?
Thanks for any help.