[Math] Prove that every idempotent element is not nilpotent element.

abstract-algebraidempotents

Let $R$ be a ring. Prove that every idempotent element is not nilpotent element.

I've got a problem with proving this question. I was be grateful, if somebody would be help me.

Best Answer

Definitions: if $x\in R$ and $R$ is a ring, then we say that an element is idempotent if $x^2 = x$, and nilpotent if for some $n\in\mathbb N: x^n=0$.

Now, suppose that $x$ is non-zero and idempotent. Note that $x^3 = x x^2 = xx = x^2 = x$. What can we say in general about $x^n$?

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