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Suppose R is a commutative ring with 1, I $\subset R$ is an ideal.

We have R-Modules A, B and C with C being flat, as well as a short exact sequence

$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$

Consider the induced sequence

$0 \rightarrow A/IA \rightarrow B/IB\rightarrow C/IC\rightarrow 0$

How do I prove that this sequence is exact? I have no idea how the flatness of C could come into play, or to be more specific, how I can use the exactness of $C\otimes\_$ (this is the only definition of flatness we have so far).

Any advice in the right direction would be appreciated.

3 Answers 3