Problem (Rotman)
Definition
Partial proof
($\Rightarrow$) Assume that $E$ is an essential extension and let $e \in E$ be nonzero. There exists an $\alpha : M \hookrightarrow E$ such that $re \in \langle e \rangle \cap \alpha(M)$ for some $r \in R$. So there exists an $m \in M$ such that $re = \alpha(m)$.
($\Leftarrow$) Already solved, don't have any questions.
Question
Under Rotman's definition of essential extension, we can't assume that $\alpha$ is the inclusion or that $\alpha(M) \subset M$. So it's possible that $\alpha(m) \notin M$. I know there's alternative definitions of essential extension that only deal with submodules $M \subset E$, but the definition here is more general and doesn't help.