Let $M$ be an $R$-module and $N_{1},N_{2}$ be submodules of $M$ such that $M = N_{1}\oplus N_{2}$. Then $M/N_{1}\simeq N_{2}$ and $M/N_{2}\simeq N_{1}$.
My Attempt:
Define a $R$-module Homomorphism $\varphi:M\rightarrow N_{2}$ given by $\varphi ((n_{1},n_{2}))=n_{2}$.
Clearly, $\varphi$ is a surjective $R$-module Homomorphism and kernel of $\varphi$ is $N_{1}$ because for all elements of the form $(n_{1},0)$ have their image $0$ under $\varphi$.
Therefore, by First Isomorphism Theorem for Modules, We have $$M/N_{1}\simeq N_{2}$$
Can anyone spot my mistake and please give me hints to correct it and solve the question?