If $M/N$ is isomorphic to $M/N'$, can we conclude that $N$ is isomorphic to $N'$?

Question

Let $N$ and $N'$ be submodules of $R-$module $M$. If $M/N$ is isomorphic to $M/N'$, can we conclude that $N$ is isomorphic to $N'$?

I believe the answer is no but I can't find a counter example. Help me some hints.

Thank you in advance.

Answer 1

Yes, it may even happen that $N'=0$, i.e. that $M$ is isomorphic to $M/N$ but $N \neq 0$. Take $M=R \oplus R \oplus \dotsc$ and $N = R \oplus 0 \oplus 0 \oplus \dotsc$.

Answer 2

Here's one (class of) example(s).

Let us restrict our attention to $\mathbb Z$-modules, i.e. abelian groups. Note that two abelian groups of order $p$ are always isomorphic, thus it is sufficient to find a group $P$ of order $p^i$ (for any $i$) with two subgroups $A$ and $B$ of order $p^{i-1}$ such that $A\not\cong B$.

For instance, if $P = \mathbf C_4 \times \mathbf C_2$, then $A = \mathbf C_4 \times 1$ and $B = \mathbf C_2 \times\mathbf C_2$ does the trick.