Why this idempotent is semicentral?

Question

Let $M$ be a $R$-module and $S=End_{R}(M)$, for any subset $X$ of $M$ let $l_{S}(X)=\{\phi \in S\space | \phi(X)=0\}$. if $l_{S}(X)=Se$ for some idempotent $e^2 =e \in S$ then how we could prove $ef= efe$ for each $f\in S$ ?

Answer 1

The proposition appears to be false as written.

Let

1. $R$ be a field $F$
2. $M=F\times F$ be thought of as column vectors, operated on by $S=M_2(F)$ on the left, and
3. $X=F\times \{0\}\subseteq M$.

Then $\ell_S(X)=\left\{\begin{bmatrix}0 & x \\ 0& y\end{bmatrix}\middle|x,y\in F\right\}=Se$ for $e=\begin{bmatrix}0&0 \\ 0 & 1\end{bmatrix}$.

But $ef\neq efe$ for $f=\begin{bmatrix}0&0 \\ 1 & 1\end{bmatrix}$.

And $fe\neq efe$ for $f=\begin{bmatrix}0&1 \\ 0 & 1\end{bmatrix}$, so it's not semicentral on that side either.