The question is from the following problem:
Let $R$ be a ring with a multiplicative identity. If $U$ is an additive subgroup of $R$ such that $ur\in U$ for all $u\in U$ and for all $r\in R$, then $U$ is said to be a right ideal of $R$. If $R$ has exactly two right ideals, which of the following must be true?
I. $R$ is commutative.
II. $R$ is a division ring.
III. $R$ is infinite.
I know the definition of every concept here. But I have no idea what is supposed to be tested here.
- Why is the ring $R$ which has exactly two right ideals special?
- What theorem does one need to solve the problem above?
Edit: According to the answers, II must be true. For III, $R$ can be a finite field according to mt_. What is the counterexample for （I） then?