Let $A\subset B$ be left $R$-submodules of $M$. Is it the case that $$\phi: A/B \oplus B \to A,\;\;\; \phi((a+B) + b) = a +b$$ is an isomorphism?
I've made a proof, but I'm not entirely convinced. In particular consider the following short exact sequence, $$0 \to G \to G'\to G'' \to 0.$$ From the first isomorphism theorem, $G \cong \text{im}\ f$ and $G'/\text{im}\ f \cong G''$, so if my claim were true it would seem that $G' \cong G'' \oplus \text{im}\ f \cong G'' \oplus G$! Since there exist non-split exact sequences, my claim musn't hold always..
It's just not true.
Consider e.g. $R=\Bbb Z$ with $A=\Bbb Z_4$, $\, B=\Bbb Z_2$. Then $A/B=\Bbb Z_2$ but $\Bbb Z_2\times \Bbb Z_2\not\cong \Bbb Z_4$.