Let be a ring with . Prove that is a division ring if and only if the only left ideals of are and . (The statement remains true when “left” is replaced by “right”.)
Suppose is a division ring. Let be a nonzero left ideal; that is, for all . Now let be nonzero. Since is a division ring, there exists such that . Then for all , , and in fact . Thus the only left ideals of are and .
Suppose the only left ideals of are and , and let be nonzero. Now the left ideal generated by is nonzero since it contains , so that . So there exists an element such that . Similarly, there exists such that . Now , so that , and hence . So is a unit in , and thus is a division ring.