Question about flat modules and exact sequences I have a basic question about exact sequences. I want to show that if I have that whenever $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ is exact then $0 \to A \otimes N \to B \otimes N \to C \otimes N \to 0$ is then $N$ is flat.
So let $\dots \to A \xrightarrow{f} B \xrightarrow{g} C  \to D \dots$ be an exact sequence. Now I want to split it into short exact sequences so I can apply what I have: $0 \to A / \operatorname{Ker}{f} \to B \to \operatorname{Im}{g} \to 0$ is exact. 
From this I get that $ A / \operatorname{Ker}{f} \otimes N \to B\otimes N  \to \operatorname{Im}{g}\otimes N  \to 0$ is exact since $- \otimes N$ is right exact. But how do I get from that to $0 \to A \otimes N \to B\otimes N  \to C \otimes N  \to 0$? I think I can't. So how do I split the long exact sequence in a better way?

The point is that you need to show that 
$\ldots \rightarrow A\otimes N \rightarrow B \otimes N \rightarrow C \otimes N \rightarrow D\otimes N \rightarrow \ldots$ is exact at $B \otimes N$,
not that $0 \rightarrow A\otimes N \rightarrow B \otimes N \rightarrow C \otimes N \rightarrow 0$ is an exact sequence. Now, the sequence which you have shown to be exact gives the desired result.
