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Let $R$ be a ring, where $a^{3} = a$ for all $a\in R$. Prove that $R$ must be a commutative ring.

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    If this is homework, what have you tried? Otherwise, Google "x3=x commutative ring" and you'll get several solutions, including http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/herstein.2011-09-24
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    You can also take a look at this [MathOverflow question](http://mathoverflow.net/questions/32032/on-a-theorem-of-jacobson).2011-09-24
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    I remember we solved this problem in class as a fun application of Jacobson's density theorem. Somehow this seems better (if a bit overkill) than the ad hoc calculations that give the other solutions.2011-09-24
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    An exercise in Herstein's textbook *Topics in Algebra*. Herstein said that, of all the mail he got concerning that textbook, the vast majority was about this single exercise.2013-01-31
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    http://mathoverflow.net/questions/29590/a-condition-that-implies-commutativity2013-01-31

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