Let $R$ be an integrally closed integral domain with fraction field $K$. Let $L$ be a finite Galois extension of $K$ and let $\sigma_1,\dots,\sigma_n$ be the elements of $Gal(L/K)$. Let $S$ be $R$'s integral closure in $L$ and let $\mathfrak{a}$ be an ideal of $S$. Consider the following statement:
The product ideal $\sigma_1(\mathfrak{a})\dots\sigma_n(\mathfrak{a})$ is generated over $S$ by its intersection with $R$.
How general is this statement? Does it always hold? Does it hold whenever $R,S$ are Dedekind domains? Is there a counterexample even when they are Dedekind? Does it depend on the Galois group?
If $R,S$ are Dedekind domains, and if $L/K$ is either quadratic or biquadratic (i.e. if $Gal(L/K)$ is either $\mathbb{Z}/2$ or the Klein 4-group) then I have rather grisly, computation-heavy proofs of the above claim. However, it feels to me like something much more general is going on, and I wonder if I have overlooked a much cleaner argument.