Idempotent elements of Jacobson Ring

Question

Let $R$ be a Jacobson ring, namely for each element $a \in R$ there exists an integer $n$, $n＞1$ such that $a ^ n = a$. How can we prove that an idempotent element of $R$ must be a central element of $R$?

Answer 1

You can show that $R$ has no nonzero nilpotents. Indeed, if $a^n=0$ and $a^m=a$, then $a=a^{m^k}$ for any $k$, so taking $k$ large enough so that $m^k>n$ we get $a=a^{m^k}=0$.

Now, in a ring with no nonzero nilpotents, every idempotent is central. To see this, let $x\in R$ and $e=e^2$; we want to show the commutator $[e,x]$ is zero. But $[e,ex]^2=0$ (compute it!), so since there are no nilpotents, this shows $ex=exe$. Similarly $[e,xe]^2=0$, so $xe=exe$. Therefore $ex=exe=xe$ as required.