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.
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.