Let $R$ be a ring with identity such that there is a positive integer $n\geq 2$ for which $r^n=r$ for all $r\in R$. Prove $R$ is commutative.
I had proven before that If $n=2$ it is commutative as follows:
$r+s=(r+s)^2=r^2+rs+sr+r^2=r+rs+sr+s\implies 0=rs+sr\implies sr=-rs$
On the other hand $-r=(-1)r=(-1)^2r=r$.
So $sr=-rs=rs$ as desired.
I seem to be stumped even with $n=3$.
Thanks in advance, regards.
PS: This is homework.