Simple question about flat modules

Question

Everything I can find about this is stated in category theory language that I do not understand. If I have an exact sequence

$... \rightarrow A \xrightarrow[\text{}]{\text{f}} B \xrightarrow[\text{}]{\text{g}} C \xrightarrow[\text{}]{\text{}}...$

and I tensor with a flat $R$-module $M$, to get the new exact sequence

$... \rightarrow A \otimes_{R} M \xrightarrow[\text{}]{\text{}} B \otimes_{R} M \xrightarrow[\text{}]{\text{}} C \otimes_{R} M \xrightarrow[\text{}]{\text{}}...$

how are the maps corresponding to $f$ and $g$ in the new exact sequence defined?

Answer 1

Actually if you tensor with any $R$-module $M$, the map induced by $f:A\to B$ is the map $\tilde{f}:A\otimes_R M\to B\otimes_R M$ given by $\tilde{f}(a\otimes m)=f(a)\otimes m$.