Show that a ring with only trivial right ideals is either a division ring or $|R|=p$ and $R^2=\{0\}$. Why would $R$ be finite?
Let $R$ be a ring such that the only right ideals of $R$ are $(0)$ and $R$. Prove that either $R$ is a division ring or that $R$ is a ring with a prime number of elements in which $ab= 0$ for all $a,b\in R$.
I don't want the proof. I am stuck at one point. Why does $R$ have to be finite here?
Hint- any infinite group has infinitely many subgroups. If the ring operation is trivial, what are the ideals?