Isomorphism of quotient modules implies isomorphism of submodules?

Solution 1:

The implication is false for all commutative non-zero rings $A$.

Indeed, just take $M=\oplus_{i=0}^{i=\infty} A$ , $N_1=A\oplus 0\oplus0...$ and $N_2=A\oplus A\oplus 0\oplus 0...$.
Since $N_1$ is isomorphic to $A$ and $N_2$ is isomorphic to $A^2$, they are not isomorphic.
However $M/N_1$ and $M/N_2$ are isomorphic because they are both isomorphic to $M$.

[To see that $A$ and $A^2$ are not isomorphic as $A$-modules the standard trick is to reduce to the case where $A$ is a field by tensoring with $A/\mathfrak m$, where $\mathfrak m$ is some maximal ideal in $A$]