Hw12 Problem 12

Return to Homework 12, Glossary, Theorems

**Problem:** Show that a division ring contains exactly two idempotent elements.

**Solution:** Let $R$ be a division ring and let $a \in R$ be idempotent. Since $a^2 =a$, we can see that $a^2 − a = a(a − 1) = 0$. First of all, if $a=0$, then $a(a-1)=0$. Otherwise, if $a\not= 0$, then $\exists a^ {-1} \in R$ due to the fact that $R$ is a division ring. Now, we have $a −1 =e(a-1) = (a^{-1}a)(a-1) = a^{-1}[a(a-1)]=a^{-1}0=0$. This means $a −1 =0 \implies a=1$ which implies there are no zero divisors. Therefore, 0 and 1 are the only two idempotent elements.