Let $X$ and $Y$ be left modules over a ring $R$, and let $K$ be a submodule of both $X$ and $Y$ such that $X/K\cong Y/K$. Does this imply that $X\cong Y$?
In the particular case that $X$ and $Y$ are of the form $X=K\oplus A$ and $Y=K\oplus B$ for some modules $A$ and $B$, the result follows from the fact that $(K\oplus A)/K\cong A$ and $(K\oplus B)/K\cong B$. But I would like to know about the more general case.
Any help is appreciated!
(My motivation to ask this question is that I want to show that if $f:X\to Y$ is a module homomorphism, then $X\cong\textrm{Ker}(f)\oplus\textrm{Im}(f)$. This would follow from the above, since $X/\textrm{Ker}(f)\cong\textrm{Im}(f)$ and $\textrm{Im}(f)\cong(\textrm{Ker}(f)\oplus\textrm{Im}(f))/\textrm{Ker}(f)$.)