Let $M$ be a R-module and let $S$ and $T$ be R-submodules of $M$. Then: $S/(S\cap T)\simeq (S+T)/T$
To prove the above Second Isomorphism Theorem for modules, we define the following:
Let $\phi$ be the mapping $S\rightarrow(S+T)/T$ such that $\phi(s)=s+T$.
Here, $\phi$ is a $R$-homomorphism with the kernel $\phi=S\cap T$.
By the first isomorphism theorem for modules, $S/(S\cap T)\simeq(S+T)/T$.
My question is whether it is possible to define the $R$-homomorphism from the opposite direction? Say,
Let $\psi$ be the mapping $(S+T)\rightarrow S/(S\cap T)$. If possible, how would such a map be defined in this case?