TITLE: If $x^3 =x$ then $6x=0$ in a ring
QUESTION [13 upvotes]: Let $R$ be a ring with unity where
$$x^3=x,\;\;\; \forall x \in R$$
How do I prove that $$x+x+x+x+x+x=0$$

REPLY [10 votes]: Hint $\rm\,\ \forall x\!: f(x) = 0\:\Rightarrow\:\forall n\in \Bbb Z\!: f(n) = 0\ (in\  R)\:\Rightarrow\: char\, R\mid\, gcd(f(\Bbb Z))$