Let $R$ be a ring, $S$ a (left) localisable subset of $R$. I have two questions regarding the localisation of $R$ at $S$.
Firstly, under what conditions can we say that $S^{-1}R$ is flat as a right $R$-module, that is the functor $R$-mod $\to R$-mod, $M\mapsto S^{-1}R\otimes_R M$ is exact?
Secondly, if M is the free $R$-module $R^n$, is it true that the localisation $S^{-1}M=S^{-1}R\otimes_R M$ is isomorphic as a $S^{-1}R$-module to $(S^{-1}R)^n$?
In particular, I am interested in the case where $R$ is a Noetherian domain, $S$ the set of all non-zero elements, and hence $S^{-1}R$ is a division ring. But I would of course like to know the answer in full generality.
