Suppose that $R$ is a simple ring that has non-trivial idempotents. I try to prove that if an element $a$ commutes with all the idempotents, then $a$ is in the center of the ring.
If we define $[x, y] := xy - yx$, then we have $[a, e] = 0$ for each idempotent $e \in R$. I know that if $e$ is an idempotent, then $e + ex - exe$ and $e + xe - exe$ are also idempotents for all $x \in R$. Using this, I have found that $[e, [a, x]] = 0$ and $[a, [e, x]] = 0$, but I can't go any further.
If I could find an equation like $[a, x][e, x] = 0$, then I would define an ideal like $\{r \in R \vert r[e, x] = 0, \forall x \in R\}$ which would be zero or equal to the ring because $R$ is simple, and so I could prove the theorem. Do you know a proof of this or have a suggestion? I take $R$ non-unital, but if you have a suggestion for $R$ unital, it is welcome.