Let $R$ be a ring such that $R$ is a simple $R$-module. Show that $R$ is a division ring.
I have an idea for this but I would like to make sure it is correct. My idea is that $R$-submodules of $R$ are just the same as ideals in $R$. So if we take any non-zero element $r$ in $R$, then the ideal generated by $r$ must be the whole of $R$ (by simplicity) and so $r$ must be a unit and $R$ is a division ring.