Let R be a ring in which x3=x for every x in R. Prove that R is a commutative ring. This is (word-for-word) in Herstein, Topics in Algebra, Ch. 3 sec. 4, problem 19. Apparently, Herstein commented that this one problem generated more mail than the entire remainder of the book. The proof (due to Derek Holt) I found is as follows: An element x is called central if xy=yx for all y in R. Note that the central elements form a subring of R. 1. xy = 0 => yx = 0. (Proof: yx = (yx)^3 = y (xy)^2 x = 0.) 2. x^2 = x => x central. (Proof: x(y - xy) = xy - x^2y = xy-xy=0, so (by 1) (y - xy)x = 0, and yx = xyx. Similarly, (y - yx)x = 0 => x(y - yx) = 0 => xy = xyx.) 3. x^2 is central for all x in R. (by 2, because (x^2)^2 = x^4 = x^2). 4. If x^2 = nx for an integer n, then x is central. (Proof: x = x^3 = qx^2, which is central by 3.) 5. x + x^2 is central for all x in R. (Proof: By 4, because (x + x^2)^2 = 2(x + x^2).) 6. By 3 and 5, x = (x + x^2) - x^2 is central, completing the proof. This proof can the found by scrolling to the very end of this page: http://www.math.niu.edu/~rusin/known-math/99/commut_ring [Broken] I understand all the steps except for 4. How did he conclude that x is central in step 4?