# Unital ring faithful as a module over itself

How do a I see that any unital ring (not commutative, not integer) is faithful as a module over itself? I considered a unital ring $R$, then $r.1=r1=r=0$, thus $Ann(1)=(0)$. Now let $r.x=rx=0$ for $r,x\in$. Then how can I conclude $r=0$? Thanks!

That $x\in {\rm Ann}_R(R)$ means that $xr=0$ for every $r\in R$. In particular $x1=x=0$.