Hi I have the following problem:
Let $R$ be a ring and $S\subset R$ multiplicatively closed and $M$ be an $R$-module.
Show: $M$ is flat $\Leftrightarrow S^{-1}M$ is a flat $S^{-1}R$-module.
I think I can show $"\Rightarrow$" but for $"\Leftarrow"$ I don't know how to start. Can someone help me please?
The statement as you wrote it is not correct.
Consider $R=\Bbb Z$ and $S= \Bbb Z \setminus \{0\}$, then $S^{-1}R= \Bbb Q$, and clearly any $S^{-1}R$ module is flat in this case, as $\Bbb Q$ is a field. But not every $\Bbb Z$-module is flat.