Quotient of modules

Question

I' m having trouble with the following exercise:

Let $Y$ be a submodule of $X$ and $X$ a submodule of $Z$ (modules over a ring $R$). Suppose that $X/Y$ is isomorphic to $Z$. I need to conclude that $X=Z$ and $Y=\{0\}$.

Can anyone help please?

Answer 1

Consider three countable disjoint sets $A$, $B$ and $C$. Set $Y=R^{(A)}$, $X=R^{(A\cup B)}$ and $Z=R^{(A\cup B\cup C)}$ (free modules with the stated bases). Then $X/Y\cong R^{(B)}$ is a free module with a countable basis, hence isomorphic to $Z$.

You probably have a much more restricted context.