1. The problem statement, all variables and given/known data Let R be a ring and suppose there exists a positive even integer n such that x^n = x for every x in R. Show that -x = x for every x in R. 2. Relevant equations 3. The attempt at a solution I solved the case where n = 2. Let x be in R. (x+x)^2= x+x = 2x, (x+x)^2 = 4x^2 = 4x. So 4x = 2x and 2x = 0. Done. I tried using this same method when n = 4 and got nowhere.