10
$\begingroup$

If $R$ is a ring such that $x^5=x$ for all $x\in R$, is $R$ commutative?

If the answer to the above question is yes, then what is the least positive integer $k \ge 6$, such that there exists a noncommutative ring $R$ with $x^k=x$ for all $x\in R$?

  • 0
    @Jas$p$er. Not necessary, from the link by curious.2011-01-06

2 Answers 2

1

The answer is that if $R$ is a ring such that for all $x \in R$ there is an integer $n(x) > 1$ such that $x^{n(x)} = x$ then $R$ is commutative. See Herstein's "Non Commutative Rings".