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If $x^2=x$ and $x$ is non-unit then $x$ is a zero divisor in a ring $R$.

I am trying to prove the contrapositive statement, that is Suppose $x$ is not a zero divisor and trying to show that $x$ is unit. I am just using the definition for nonzero $r\in R$: $x^2r-xr=0\implies x(xr-r)=0\implies xr=r$ and eventually I get $x=1$ which says it is a unit. Is this correct. Also, is there a direct way of proving this claim?

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From $x(1-x)=0$ and $x$ nonzero divisor what you get?

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    $\begingroup$ Ahh! I missed such a simple observation $\endgroup$
    – Rutherford Mark
    Nov 17, 2014 at 21:05
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    $\begingroup$ @RutherfordMark You can also think directly: if $x$ isn't a unit, then $x\ne 1$, that is $1-x\ne 0$ and from $x(1-x)=0$ ... $\endgroup$
    – user26857
    Nov 17, 2014 at 21:07

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