Quotient different modules by the same module to get the same module.

Question

More precisely, If i have modules $M$ and $M'$ and $N$ and $N'$ does $M/N \cong M'/N'$ and $N \cong N'$ imply that $M \cong M'$?

What if we also know that $N$ is free or that $M/N$ is free?

Answer 1

In general the answer is no, even if $N$ is free. For example take $N=N'=\mathbb Z$ and $M/N=M'/N'=\mathbb Z/2\mathbb Z$ as $\mathbb Z$-modules. Then you can take $M=\mathbb Z$ and $M'=\mathbb Z/2\mathbb Z\times \mathbb Z$, the map $N\to M$ being multiplication by $2$ and the map $N'\to M'$ being injection to the second factor.

If $M/N$ is free I believe the answer is yes, since free modules are projective and so the exact sequence $N\to M\to M/N$ splits.