If $R$ is generated by idempotents, then $\text{Ann}(R)=0$?

Question

Let $R$ be a ring (not necessarily commutative or unital) which is generated by idempotents. I'd like to know if $\text{Ann}(R)=0$ must holds. Here I use $\text{Ann}(R)$ to denote the set of all elements $r\in R$ such that $rR=Rr=0$. All I knew is that it holds when $R$ is commutative.