On idempotent elements that are contained in center of a ring Let $e$ and $f$ be idempotent elements of a ring $R$. Assume that $e,f$ are contained in center of $R$. Show that $Re=Rf$ if and only if $e=f$ 
Help me a hint to prove it.
Thank in advanced.

For $r\in R$, we have $r\in Re$ if and only if $re=r$. 
Indeed $re=r$ implies $r=re\in Re$; conversely, if $r\in Re$, then $r=xe$ for some $x$, and so $re=xee=xe=r$.
From $Re=Rf$ we get $f\in Re$, so $f=fe$. For the same reason $e=ef$. But since $e$ and $f$ commute, we have $e=f$.
