I'm trying to prove the next proposition about modules, where $L,$ $M$ and $N$ are modules over a $R$ ring:
- If $L \leq N \leq M$ and $L \leq_{e} M$ then $N \leq_{e} M$
- Let $f:M \rightarrow L$ be a homomorphism and $ L' \leq_{e} L $ then $f^{-1}[L'] \leq_{e} M.$
Where $L \leq N$ means that $L$ is a submodule of $N$ and $L\leq_{e}M$ means that there exist a monomorphism $\alpha : L \rightarrow M$ such that, for every $K\leq M$ with $K \neq \{0\}$ occur $K\cap \alpha (L) \neq \{0\}$
I've tried to prove the second part by taken the monomorphism $\alpha : L' \rightarrow L$ and a function $ g:f^{-1}[L']\rightarrow M $ given by $ g=\alpha \circ f $ but I think that $f$ must be an isomorphism for this to work.
Any help is appreciated, thank you.