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If in a ring $R$, $x^2=x$ for all $x$ then show that $2x=0$ and $x+y=0$ implies that $x=y$

Qurultay
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He Er Khan
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  • If $x^2=x$ for all $x\in R$, then for $x,y\in R$ we have also $(x+x)^2=x+x$, or $$x^2+xx+xx+x^2=x+x\implies x+x+x+x=x+x $$ thus $x+x=2x=0$ – Qurultay Feb 05 '20 at 10:27
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    Moreover, if $x+y=0$, then $x=x+0=x+2y=x+y+y=0+y=y$. – Will Feb 05 '20 at 10:32
  • Welcome to MSE. In order to get responses that suit your needs, please include in the body of the question your own thoughts, the effort made so far, and the specific difficulties that got you stuck. BTW, I didn't downvote. – Lee David Chung Lin Feb 05 '20 at 10:43

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