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A ring $R$ is a Boolean ring if $x^2=x$ for all $x\in R$. By Stone representation theorem a Boolean ring is isomorphic to a subring of the ring of power set of some set.

My question is what is an example of a ring $R$ with $x^3=x$ for all $x\in R$ that is not a Boolean ring? (Obviously every Boolean ring satisfies this condition.)

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    Would $\mathbb{Z}_3$ work?2011-01-06
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    You are right. I should have thought about that before I ask. Note that the same question can be asked for $x^n=x, n\ge 4$.2011-01-06

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