Let $R$ be a ring, where $a^{3} = a$ for all $a\in R$. Prove that $R$ must be a commutative ring.
If $a^3 =a$ for all $a$ in a ring $R$, then $R$ is commutative.
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abstract-algebra
ring-theory
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8If 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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2You 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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0I 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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4An 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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4http://mathoverflow.net/questions/29590/a-condition-that-implies-commutativity – 2013-01-31