If $x^5 = x$ in a ring, is the ring commutative? [duplicate]

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 $x\in R$?

Solution 1:

https://mathoverflow.net/questions/29590/a-condition-that-implies-commutativity

https://mathoverflow.net/questions/32032/on-a-theorem-of-jacobson

Solution 2:

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".