A question regarding a central idempotent in a ring $R$

Here's kind of a slick solution (don't read further if you want to figure it out for yourself!)

$$ab = 1$$ $$\Longrightarrow (ba)^{2} = b(ab)a = ba$$

So $ba$ is an idempotent and therefore central. So we have:

$$b(ba) = (ba)b$$ $$\Rightarrow ab(ba) = a(ba)b$$ $$\Rightarrow (ab)(ba) = (ab)(ab)$$ $$\Rightarrow ba = 1$$