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:

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> The product ideal $\sigma\_1(\mathfrak{a})\dots\sigma\_n(\mathfrak{a})$ is generated over $S$ by its intersection with $R$.
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**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.