Show that a ring with only trivial right ideals is either a division ring or $|R|=p$ and $R^2=\{0\}$.

I have trouble with the following problem: Let $R$ be a ring (it doesn't necessarily have multiplicative identity). If the only left Ideals of $R$ are $(0)$ and itself then $R$ is a division ring or $|R|=p$ for some prime $p$ and $ab=0$ for all $a,b\in R$. I tried to prove that $R$ is a domain, because if that's true then I have an easy way of proving the exercise, but I don't know if it's true that $R$ is a domain

Solution 1:

As per this solution, it is a division ring if $R^2\neq\{0\}$.

If $ab=0$ for every $a,b\in R$, then it is a commutative ring, and the ideals are exactly the abelian subgroups of the underlying group.

Then ask yourself what simple abelian groups look like.