Reading the class notes I stumbled upon an unproved lemma that I am having issues proving.
The lemma is, if $R$ is a ring with no non-zero nilpotent elements and $e$ is idempotent then $e$ is central.
In other words, if $e$ is idempotent in such a ring, then $ex=xe$ for all $x\in R$.
I am however not sure how to actually prove the statement.
So far I have been playing around trying to find polynomials that can be factored into linear identical factors (e.g (x+e)^2) to see if I can make them be zero by converting an $e^k$ coefficient into $e$.
But something isn't fully clicking about this lemma.
Consider $a=ex-exe$ and $b=xe-exe$. Then $a^2=b^2=0$.
Since there are no nilpotent element, $ex-exe=xe-exe=0$ and thus $xe=ex$.