Q:

Non-zero idempotent element is not nilpotent

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.

A:

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.

