[Math] Showing that the only idempotents in $R$ are zero and one

Question

I have the following question that I have to solve however I cannot achieve.

Let $R$ be a ring with $1$ and suppose $R$ has no zero divisors. Show that the only idempotents in $R$ are $0$ and $1$.

Please help me to solve this question.

Your prompt reply will be greatly appreciated.

Thanks in advance.

Answer 1

Suppose $e$ is idempotent. Then $e^2=e$, so $e^2-e=0$. Trying factoring and see what happens, using the fact that there are no zero divisors.