If $K/k$ is a finite Galois extension of fields, with Galois group $G$, there's an isomorphism $ K \ \otimes_k \ K \simeq \oplus_{\sigma_i \in G} \ K$ given by sending $a \otimes b$ to $ (..., \sigma_i(a) b, ...)$ and extending linearly. This is even an isomorphism of $K$-vector spaces, where $a \in K$ is acting by multiplication on the first factor in $K \otimes K$, and by multiplication by $(\sigma_i(a))$ on $\oplus_{\sigma_i} K$. There's also an obvious $k[G]$-module structure that's preserved.
My question is what happens if instead of $K$, we consider $R \otimes_S R$, where $R$ and $S$ are the rings of integers of number fields $K$ and $k$ respectively. I tried playing around with quadratic extensions, and I'm pretty sure that the same map above from $R \otimes R$ to $\oplus R$ is not surjective, so the same argument doesn't work? Is there a nice description of $R \otimes R$ as either an $R$ module, or an $S[G]$ module?
Thanks.