Let $M$ be a left $R$-module having no proper essential extension and which is embedded in an injective left $R$-module $I$. Let $S$ be a complement (in $I$) of $M$, namely, a submodule of $I$ maximal with $S∩M=0$. One could deduce that the image of $M$ in $I/S$ is essential in $I/S$. Could we infer that the image of $M$ in $I/S$ equals $I/S$? And how?
Thanks for help!
I think you could infer that: since $M\longrightarrow I/S$ is injective, the image of $M$ in $I/S$ is isomorphic to $M$.