[Math] Non-zero idempotent element is not nilpotent

Question

I have a problem that I need to solve but I have trouble in solving the following question.

Question is;

Let $a \in R$ be a nonzero idempotent. Show that $a$ is not nilpotent. ($R$ is a ring)

I will appreciate your help.

Thanks in advance.

Answer 1

Since $a$ is an idempotent element then $$a^2=a$$ hence we have $$\forall n\in \Bbb N,\qquad a^n=a\ne0$$ hence $a$ isn't nilpotent.