Let R be a ring with identity and not necessarily commutative. Let $M_1, M_2$ be left $R$-modules with submodules $S_1, S_2$ respectively such that $M_1/S_1 \cong M_2$ and $M_2/S_2 \cong M_1.$ Is it necessarily true that $M_1\cong M_2$ ? What if we also assume the modules are finitely generated?
It is true in the vector space case when $R$ is a field, but I doubt it remains true generally. Can someone help me find a counterexample?
You ask if epi-equivalent modules are isomorphic. You find sufficient conditions in the paper Correct classes of modules by R. Wisbauer. But it fails in general, for example $R/I \oplus R^{\oplus \mathbb{N}}$ has an epimorphism to $R^{\oplus \mathbb{N}}$, which has an epimorphism to $R/I \oplus R^{\oplus \mathbb{N}^+} \cong R/I \oplus R^{\oplus \mathbb{N}}$, but (usually) they are not isomorphic.