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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!

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

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  • $\begingroup$ Oh, I see. Thank you! $\endgroup$
    – user221151
    Apr 4, 2015 at 0:09

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