TITLE: Commutativity characterization?
QUESTION [9 upvotes]: Let $R$ be a ring (not necessarily unital) and for any $x\in R$ there is an integer $n \geq 2$ s.t. $x=x^2+\cdots+x^n.$ Does it imply that $R$ is commutative?

REPLY [6 votes]: Many of these questions are particular cases of the following general theorem of Herstein: 

If $R$ is a ring with center $Z$ and for every $a\in R$ there exists a polynomial $p_a\in\mathbb Z[X]$ such that $a^2p_a(a)-a\in Z$, then $R$ is commutative.

In your case just take $p_a(X)=1+X+\cdots+X^{n-2}$ for each $a\in R$ and conclude that $R$ is commutative.