Show that a division ring R contains exactly two idempotent

Show that a division ring R contains exactly two idempotent elements.

Solution

Let a be an idempotent element in a division ring.

Then a ² = a, so a ² a = 0 which implies that a(a 1) = 0.

Since a division ring has no zero divisors either a = 0 or or a = 1.

Hence, the idempotent elements of a division ring are exactly 0 and 1.