Identifying a quotient module with a submodule

Question

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: $M\subset R$.

Answer 1

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.

Answer 2

I think in general the answer is no.

Think about the example of $R=\mathbb{Z}$ and let $M=R$ as an $R$ module. Since $R$ is a PID, all ideals of $\mathbb{Z}$ (in this case submodule of $M$) are $n\mathbb{Z}$,isomorphic to $\mathbb{Z}$.

But all the nontrivial quotients of $\mathbb{Z}$ are like $\mathbb{Z}/n\mathbb{Z}$. They are not free.

Edit: for the last line, exclude case $n=1$ or $-1$ or $0$.