Examples of Abelian rings

Question

Let $R$ be a ring. an element $e$ in $R$ is idempotent if $e^2=e$. we say that $R$ is Abelian if for each $x \in R$ and each idempotent $e \in R$ we have $ex=xe$. $R$ is commutative if for each $a,b \in R$ we have $ab=ba$. is there any example of the Abelian ring that is not commutative?

Answer 1

Yes, there is. Take any non-commutative reduced ring. In reduced rings idempotents are central.

Answer 2

In counterpoint to noncommutative reduced rings, you also have noncommutative local rings. The only idempotents there are $0,1$ which are always central, but here you can have nilpotent elements.

It also turns out there is an example due to Chase that is neither local nor reduced. If the link ever breaks you can find it here:

T.Y. Lam. Lectures on modules and rings. (2012) pp 47-48