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Let $R$ be a ring with unity where

$x^3=x,\;\;\; \forall x \in R$

How do I prove that $x+x+x+x+x+x=0$

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    So if $x^n=x$, then multiplication by $(m^n-m)$, with $m$ natural, annihilates the ring element. In this case you you present $6=2^3-2=0$.2013-07-31

3 Answers 3

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Simply calculate $2 = 1+1 = (1+1)^3 = 8$ so $6 = 0$ (here I use an integer $n$ to mean the unit added to itself $n$ times). Now note that any element added to itself $6$ times is the same as $6$ times that element, which is then $0$.

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    @Temitope: You have only one tool to use: "$x^3 = x$". So you simply plug in the simplest things you know to get more information out of the tool. $2$ is about the fourth simplest thing you know. $2x$ is about the fifth simplest thing you know if you need something that isn't constant.2012-12-12
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Hint $\rm\,\ \forall x\!: f(x) = 0\:\Rightarrow\:\forall n\in \Bbb Z\!: f(n) = 0\ (in\ R)\:\Rightarrow\: char\, R\mid\, gcd(f(\Bbb Z))$

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    @Thomas Yes, I've clarified that, thanks.2012-12-12
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$(x + x)^3 = x^3 + 3x^3 + 3x^3 + x^3$ by the binomial theorem. Now use the condition that $x^3 = x$ for all elements in the ring to conclude that $2x = 8x$, from which the desired conclusion follows.