Isomorphic Submodules implies Isomorphic Quotients?

Question

I am trying to solve this exercise in the Book of Roman: if $S$ and $T$ are isomorphic submodules of a module $M$ it does not necessarily follow that $M/S\approx M/T$. Prove that this statement does hold if all modules are free and have finite rank.

I have a counterexample but not a proof.

Answer 1

We have $\textrm{rank}\left(M\right)=\textrm{rank}\left(S\right)+\textrm{rank}\left(M/S\right)$ and $\textrm{rank}\left(M\right)=\textrm{rank}\left(T\right)+\textrm{rank}\left(M/T\right)$. Since $S\approx T$ then $\textrm{rank}\left(M/S\right)=\textrm{rank}\left(M/T\right)$ . Therefore $M/S\approx M/T$ .