Question

An element of R is called idempotent if a ²=a. Show that a division ring contains exactly 2 idempotent elements.

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Answer 1

Suppose $a\in R$ is such that $a^2=a$. We have two possibilities :

• $a=0$, as $0\cdot 0 = 0$,
• $a\neq 0$, in which case there exists $a^{-1}$ and thus we have $id=a\cdot a^{-1}=(a\cdot a)\cdot a^{-1} = a\cdot id = a$, so $a=id$.

In addition, as in a ring we have $id \neq 0$ by definition, then the set of idempotent elements consists of 2 elements : and $id$.