If R is a ring, and has the property that x^3=x for all x in R, then show R is commutative. Generalize for x^k=x, k an integer >3.
I can do this for x^2=x. First use x^2+x^2=x+x=(x+x)^2 = 4x^2 (abusing notation a little) which means x^2+x^2=x+x=0, ie, x=-x for all x in R. Then...