$R$ is a ring with identity.If $\ \pi\in R\ $ is an idempotent element ($\pi^2=\pi$),How to prove $\operatorname{ann}_\text{l}(\pi)=R(1-\pi)$? where $\operatorname{ann}_\text{l}(\pi)$ represents the left annihilator of $\ \pi\ $.
I know how to prove $R(1-\pi)\subset\operatorname{ann}_\text{l}(\pi)$.i want to prove $\operatorname{ann}_\text{l}(\pi)\subset R(1-\pi)$ to complete the proof.
$\alpha\in \operatorname{ann}_\text{l}(\pi)\Rightarrow\alpha\pi=0$. And I want to prove $\alpha=r(1-\pi)$ for some $r\in R$. But I have no idea, can someone give me some hints? Thank you.