If I have a commutative ring $R$ and an exact sequence
$0\to M'\to M\to M''\to 0$ where $\epsilon:M'\to M$ and $\sigma:M\to M''$
do I get an exact sequence
$0\to M'\to M\to M''\to 0$ by means of $\epsilon \circ id:M'\times N\to M\times N$ and $\sigma\circ id:M\times N\to M''\times N$?
By $f\circ g$ I mean the mapping $(x,y)$ to $(f(x),g(y))$ (not sure how to do the tensor product symbol).
The reason I ask is that my notes are a scrambled scrawling of material I cannot make sense of. And It looks like I have a lemma without a conclusion here but this is what I Guess it is. Can anyone confirm? Thanks so much.