Let $R$ be an arbitrary ring (not necessarily commutative etc). Let $M$ be a left $R$-module and $N\subset M$ a submodule of $M$. Consider the quotient $M/N$, is there a natural way to identify via an isomorphism (of $R$-modules), say, the quotient $M/N$ to a submodule $M_{1}\subset M$ of $M$?
EDIT: $M\subset R$.
Best Answer
The answer is no in general.
Take as a counter-example $R=M=\mathbf Z$, and $N$ a non-trivial ideal $n\mathbf Z$ ($n\ne 0,\pm1$). Then $\mathbf Z/n\mathbf Z$ is a torsion module, whereas any submodule of $\mathbf Z$, i.e. any ideal of $\mathbf Z$, is free.